Ramakrishna was a great Sage. So which school of thought did he belong or followed.
The great mystic & Saint, Sri Ramakrishnan Paramahamsa, born in Bengal, started his journey with unflinching devotion to Maa KAli which later also took him towards the Advaita School of Vedanta, after the incident - " when he cut Maa Kali into two pieces using the sword of jnana (knowledge) ".
His Wikipedia Page quotes as follows
Sri Ramakrishna experienced spiritual ecstasies from a young age, and was influenced by several religious traditions, including devotion toward the Goddess Kali, Tantra, Bhakti and Advaita Vedanta.
Vivekananda portrayed Ramakrishna as an Advaita Vedantin.
The message of Sri Ramakrishna to the modern world, which he gave through his life and through his recorded conversations, may be briefly stated as follows:
The goal of human life is the realization of the Ultimate Reality which alone can give man supreme fulfilment and everlasting peace. This is the essence of all religions.
The Ultimate Reality is one; but it is personal as well as impersonal, and is indicated by different names (such as God, Ishvar, etc) in different religions.
Further, in his Book - "Ramakrishna Paramahamsa: Sadhaka of Dakshineswar" ,
Amiya P Sen observes:
It is difficult to be precise or categorical while placing Sri Paramahamsa within the complex range of Vedantic thoughts. There is good reason to believe that he was dissatisfied with the advaitic tendency to collapse the distinction between God and man. To illustrate this, he would employ a metaphor attributed to the Sakta-Tantric poet, Ramoprasad Sen. It is said that Ramoprasad was keen to taste the sweetness of sugar, without turning into sugar; that is, he preferred to remain in a state of dualistic bhakti rather than be drawn into a state of abstraction without identity. Sri Ramakrishnan's ecstatic attachment to the goddess KAli, his recurrent and potent use of devotional songs, his visions of various gods and goddesses, when in saadhnaa, all point to the nature of a bhakta (devotee).
On the other hand, after being initiated into sanayaasa, by a monk of Sankarite Dasanami order, he also chose to stay in a non-dualist bhava (mood) for a period of six months. Importantly too, his reading a text like Bhagavad Gita suggest not bhakti (devotion) but tyaga (renunciation), an attitude more easily identifiable with a practitioner of Jnan Marga (the Knowledge path).
Some people also give terms like "neo-vedantin", but I personally don't understand or comprehend it, whatever it might connotes.
To conclude , with this QnA discussion, and a brief reference from here and what we have discussed above, one may be able to arrive at some conclusion that - it's really not feasible nor tenable to limit Sri Ramakrishnan Paramahansa with any particular or specific ideology systems or School of thought. He was a great Hindu saint, that too, a Paramahamsa at that, whose teachings, may be accommodated and understood as jewels of universal prajna (wisdom) of truth and reconciliation with the Universal Reality we call Brahman.
He followed all schools Vedanta thus unifying all aspects of Hinduism. He even followed other religions even became a muslim and christian . If one goes through authentic biographies of Ramakrishna such as Ramakrishna Kathamrita and Ramakrishna Leelaprasanga, one can come to conclusion of the message of unity of Hinduism and other religions as revealed by Ramakrishna's various sadhnas(which are not limited to Hinduism).
Similar question (What is Sri Ramakrishna's philosphy) was asked by Swami Sharvananda to Swami Turiyananda.
Swami Turiyananda was a direct monastic disciple of Sri Ramakrishna and a knower of Brahman. He was such an austere soul that Sri Ramakrishna, looked upon him as the embodiment of that renunciation which is taught in the Bhagavad Gita.
I'm reproducing the letter of Swami Turiyananda from the book : Spiritual Treasures, in which he tries to answer the question.
Alternatively you can watch a video here where Swami Sarvapriyananda - Minister Incharege of Vedanta Society of New York, explains similar question.
I received your letter some time ago, but I could not write to you until now because I was ill. Though the subject [What is Sri Ramakrishna's philosphy] is extremely difficult, with the Lord's grace, I will try my best to answer your questions.
It is not so easy to speak about Sri Ramakrishna's philosophy. It seems to me that to encourage the followers of all religions, he declared "As many faiths, so many paths." He made this statement after he himself had practised the disciplines of various religious paths and experienced that their paths lead to the same Truth.
The ultimate Truth is one and nondual. It is called by various names: Brahman, Paramatman, Bhagavan [Lord], God, and so on. Whoever has realized it has tried to express it according to his own temperament and understanding by giving it a particular name. But nobody has been able to express the whole truth. "What he is, he is" - that is the final conclusion of those who have realized him.
From different standpoints, Gaudapada's doctrine of no creation, Shankara's doctrine of superimposition, Ramanuja's doctrine of transformation, and [Sri Kantha's] doctrine of Shivadvita - each one of these is true. Again, apart from all these doctrines, God is beyond all human expression and beyond the cognition of the mind. The founders of all these philophical systems practised austerities, and having, received God's grace, they preached the various doctrines at his command. God is the subject from which these doctrines evolved, but he himself is beyond them. The philosophy of Sri Ramakrishna is to express this truth. That is what I think.
Hanuman said to Rama: "O Lord, when I identify myself with the body, I am your servant. When I consider myself as an individual soul, I am a part of you. And when I look upon myself as the Atman, I am one with you - this is my firm conviction." Sri Ramakrishna referred to this statement as the best conclusion of different phases of spiritual exprience.
Why should it not be possible to see the worship vessels as Brahman, saturated with consciousness? [This is a reference to Sri Ramakrishna's vision in the Kali Temple, where it was revealed to him that everything is Pure Spirit.] "Throughout the universe he exists, pervading every being and every thing, animate and inanimate." There is nothing but God. God verily is all. Because we cannot see him, we see objects instead, but the fact is that God is everything. Names and forms originate from him and remain in him. The waves, the foam, the bubbles - they are all nothing but water. Who cares if your doctrine of superimposition stands or falls! He who has known this truth [that Brahman is all] cannot be content with a lesser viewpoint.
The Master used to experience a state of beyond all thoughts and ideas. That state transcends name and form, words and mind. There exists only One without a second, beyond the realm of prakriti [that is, beyond relativity]. Where is the doctrine of superimposition or the doctrine of no creation in that realm of Oneness? And yet again, all doctines - whether of superimposition, or of no creation, or of transformation, etc. - originate from the Lord.
He alone is the Reality, the Truth. And again he is the source of all individual beings and of the universe. This manifestation is also true if he is not forgotten. Name and form become unreal if we forget God because they cannot exist without him. [Sri Ramakrishna said: "Zeros added together amount to zero. Place the digit one before them, and they add in value." The digit one is God.] But if he dwells in our thoughts, only then can we understand the truth that "the pith belongs to the sheaths and the sheaths belong to the pith." [A reference to Sri Ramakrishna's saying that as long as the plantain tree contains sheaths, it also contains pith. He was illustrating the point that while God keeps the "ego of a devotee" in a person, the Relative (the Sheaths) is real and so is the Absolute (the pith).] At that time one can understand these sayings of the Gita: "All things in this universe are pervaded by Me" [IX.4]; "All is strung on Me as a row of gems on a thread." [VII.7]
The main thing is we must see God. When we see the Lord, everything else disappears. One experiences God as everything. Before we see him we have doubts and confustion and all sorts of theories and controversies. But these cease to exist as soon as we see him. Then one experiences uninterrupted peace and bliss.
Therefore, Sri Ramakrishna's philosophy is: In whatever way and at any cost we must attain God. The Master said, "Tie the nondual knowledge in the corner of your cloth and then do as you please." This means: Once you attain God, it does not matter which doctrine your temperament bids you to uphold. Liberation is assured when you know him. Then there is no more bondage. After death, whether you take another body or not depends upon your wish.
The seekers of nirvana [final liberation] consider this world a dream. They merge their minds in the impersonal aspect of Brahman and become one with it. The devotees who are attached to the Personal God consider this world a manifestation of God's power. They attach themselves to the Lord, who is Existence-Knowledge-Bliss Absolute. They are not afraid to be born again and again. They consider themselves to be playmates of God, and they come to the world to join his divine play. They delight in the Atman and at the same time remain devoted to God. They covet nothing in this world. They even refuse to accept nirvana if it is offered to them. That is enough for today.
"From the book : Spiritual Treasures: Letters Of Swami Turiyananda", translated and edited by Swami Chetanananda, letter 115, p.195-198
Sri Ramakrishna has hinted that there is some truth in all schools. He did not belong to any but accepted the truth of all while not accepting the faults of all these sects.
"Listen to a story. Once a man entered a wood and saw a small animal on a tree. He came back and told another man that he had seen a creature of a beautiful red color on a certain tree. The second man replied: 'When I went into the wood, I also saw that animal. But why do you call it red? It is green.' Another man who was present contradicted them both and insisted that it was yellow. Presently others arrived and contended that it was grey, violet, blue and so forth and so on. At last they started quarrelling among themselves. To settle the dispute they all went to the tree. They saw a man sitting under it. On being asked, he replied, 'Yes, I live under this tree and I know the animal very well. All your descriptions are true. Sometimes it appears red, sometimes yellow, and at other times blue, violet, grey, and so forth. It is a chameleon. And sometimes it has no color at all. Now it has a color and now it has none. 'In like manner, one who constantly thinks of God can know His real nature; he alone knows that God reveals Himself to seekers in various forms and aspects. God has attributes; then again He has none. Only the man who lives under the tree knows that the chameleon can appear in various colors, and he knows, further, that the animal at times has no colors at all. It is the others who suffer from the agony of futile arguments........ God reveals Himself in the form which His devotee loves most."
The Gospel of Sri Ramakrishna, The Master with the Brahmo Devotees (I)