I'll cite from scriptures to show why Goddess Saraswati is considered as the Goddess of knowledge,wisdom,learning,eloquence,memory , poetic skills etc.
1) The Saraswati Suktam found in Rig Veda 1.3.10 has mentioned(in the last verse of the Suktam) her as the Goddess of learning and eloquence.
2) Next is the Saraswati Kavacham found in the Brahma Vaivarta Purana.Lord Brahma is the Risi(Seer) of this Kavacha.
In this Kavacha Goddess Saraswati is identified with the Goddess of speech(Vagdevi),Goddess of Knowledge(Vidyaadhistatri,Vidya Swarupi),One who resides in prose and poetry(Gadya Padya Nivasini) and one who resides as Vidya in sacred books.
In Phalasruti too, it is stated that recitation of this Kavacha yields memory power,mental sharpness and immense knowledge.
3)The last one is another Stuti of Devi Saraswati done by Rishi Yajnavalkya also to be found in the Brahma Vaivarta Purna.
In this stuti too,Goddess saraswati is depicted as the bestower of Jnana(wisdom),Smriti(memory),Vidya(knowledge) & Kavitwa Shakti(the power of poetry).
She is also depicted as the "Adhistatri of all Vidyas" and as "Akshara Swarupa"(the Personification of alphabets).
She is depicted as the embodiment of Smriti Shakti,Jnana Shakti,Buddhi Shakti and Kalpana Shakti and also as the ruler of tongue,mind,thought,speech and anything to do with brains.
Conclusion-Goddess Saraswati is considered as the Goddess of Knowledge due to facts that are firmly rooted in our scriptures and not because of beliefs.
That Goddess Saraswati is to be worshipped on Sukla Panchami Tithi of the Magha month is mentioned in scriptures.
Mentioned in other Puranas as well but here i'm quoting from the Devi Bhagavatam as found on this page:
Sage Narayana described the methodology of worship of Devi Sarasvati,
the powerful manifestation of Mula Prakriti, the Goddess of Knowledge,
Mental Faculties, Memory, ‘Mantras’, Literature and Fine Arts. Her
symbol is ‘Veena’ (Lute) and Books; Her ‘Vahana’(Vehicle) is ‘Hamsa’
(Swan) and Her male counterpart is Para Brahma. On every Sukla
Panchami of Magha month and on the day of commencing education, the
worship of Sarsvati is to be performed as per Kanva Sakha Yajur Veda
hymns, if possible.