Yes, Lord Ganesha is a form of the Eternal Brahman. Therefore He is eternal. The stories relating to the birth of Lord Ganesha are also of a deep spiritual significance. Here is what Veda and some saints have said about Lord Ganesha.
Symbolic meaning of Lord Ganesha
Lord Ganesha is a form of the eternal Brahman. The form and stories of Lord Ganesha has a deep spiritual significance.
This is what Sri Swami Sivananda says about Lord Ganesha
Lord Ganesha is an embodiment of wisdom and bliss. He is the Lord of Brahmacharins. He is the foremost among Brahmacharins or celibates.
He rides on the Vahana, the small mouse. He is the Presiding Deity of the Muladhara Chakra.
He is the Lord who removes all obstacles in the spiritual path and brings worldly success. So He is called Vighna Vinayaka. His Bijakshara is Gang. He is the Lord of harmony and peace.
Lord Ganesha represents OM or the Pranava. Pranava is the chief Mantra of the Hindus. Nothing can be done without uttering it. Hence the practice of invoking Lord Ganesha before beginning any rite or work. The two feet are His Jnana Sakti and Kriya Sakti. Lord Ganesha has the elephant-head as that is the one figure in nature which is of the form of Pranava.
Riding on the mouse represents that He has killed egoism. He holds Ankusa. This represents that He is the Ruler of the world. This is the emblem of Divine Royalty.
Ganesha is the first God, Adi-Deva. Mouse is a small creature. Elephant is the biggest of all animals. Riding on a mouse and wearing the head of an elephant denote that He is the Creator of all creatures, from the biggest elephant to the smallest mouse. Elephants are very wise. Wearing the head of an elephant indicates that Lord Ganesha is an embodiment of wisdom. It also denotes the process of evolution. The mouse gradually evolves into an elephant and finally becomes a man. That is the reason why Ganesha has a human body, the head of an elephant, and mouse as His vehicle. This is the symbolic philosophy of His form.
He is the Lord of Ganas or groups, such as the group of elements, the group of senses, the group of Tattwas. He is the head of the followers of Siva.
Ganesha as eternal Brahman
The Ganapathi Atharvasirsha says that Lord Ganesha is the imperishable Brahman
tvam vaangmayastvam chinmayah
tvam pratyaksham brahmaasi
tvam jnaanamayo vijnaanamayosi
You are the word, deed and thought . You are the truth, consciousness and bliss. You are the One only and non-dual Universal Self. You are the personified Brahma, appearing before us. You are the entire knowledge and wisdom.
Sri Shankaracharya on Lord Ganesha
In Sri Shankaracharya's work Ganesh Bhujangam, he says that Sri Ganesha is beginningless and Pranava.
Yamae Kaa Ksharam Nirmalam Nirvi Kalpam
Gunaa Teeta Maa Nanda Maa Kaara Shoonyam
Param Paara Mom Kaara Maan Maaya Garbham
Vathanthi Prakalpam Puraanam Tameed (Sanskrit version, English version)
I praise that Ganesha,
Who is the beginning of the universe, Who is pure,
Who is free from differences, Who is beyond the qualities,
Who is pure bliss, Who is formless, Who is beyond everything,
Who is OMkar, Who is the creator of illusions, and Whom eminent Purana sing.
Thus from advaita perspective, Lord Ganesha symbolizes the Eternal Omkara or Pranava, which is not different from Brahman Himself. All the best.