Why doesn't Gandalf completely remember who he is upon returning in The Lord of the Rings? In one of many memorable moments from The Fellowship of the Ring, Gandalf sacrifices himself against the Balrog in the Mines of Moria, allowing the other Fellowship members to escape unscathed while he tumbles into the depths, fiery demon close behind. Assumed dead, the loss of Gandalf is keenly felt among the Fellowship, but they bravely carry on regardless. When Aragorn, Legolas and Gimli venture through Fangorn Forest, however, they discover their wizard friend transformed into Gandalf the White, who resumes his role leading the fight against Sauron.
In a world where dragons hoard gold and a simple ring can bring about the apocalypse, a resurrected wizard is hardly beyond the realms of possibility. More curious than the means of Gandalf's return is how Ian McKellen's character doesn't entirely recall who he is afterwards. When revealing those shiny new robes, Gandalf the White actually announces himself as "Saruman, rather Saruman as he should have been." Then when Aragorn first uses the name "Gandalf," the figure replies, "Gandalf? Yes... that was what they used to call me... that was my name" before finally christening himself "Gandalf the White." The dialogue is different in J.R.R. Tolkien's source material, but the overall emphasis and Gandalf's hazy memory remains.
Gandalf doesn't just feel detached from his own name because he caught a bump on the head while fighting the Balrog - it's a sign that Gandalf the White and Gandalf the Grey are almost two separate people. Though his explanation is vague, Gandalf the Grey does die in the process of defeating the Balrog. By the grace of Tolkien's God, Eru, Gandalf is permitted to return in a completely different form, so while the character is still Gandalf, only his spirit remains from before. The separation is much clearer in the Lord of the Rings books, where Gandalf says he'll let Aragorn and the others continue using that name, as if it doesn't really belong to this new fellow in white at all.
The stark change from Gandalf the Grey to Gandalf the White is addressed by Tolkien in his published letters, where the author comments on Gandalf's drastic evolution by saying, "of course he remains similar in personality and idiosyncrasy, but both his wisdom and power are much greater." Gandalf's change isn't entirely dissimilar to The Doctor in Doctor Who - every regeneration feels like their past memories belong to someone else.
Additionally, "Gandalf" is a name spoken only in Middle-earth. The mysterious old Hobbit-botherer was originally an angelic spirit from the Undying Lands, chosen to aid in the battle against Sauron along with several others. These lucky Maiar were given physical forms and became the wizards of Middle-earth, who only then took names such as Gandalf, Saruman and Radagast. When Gandalf the White describes his death in The Two Towers, he claims to have "strayed out of thought and time" where "every day was as long as a life age of the Earth." Though only days have passed in Middle-earth, no one has used the name "Gandalf" for a very long time, from his perspective. If anything, Gandalf would've reverted to Olórin during this period, though he most likely wasn't referred to by any name at all. In either case, it's hardly surprising that the word "Gandalf" feels unfamiliar.
The emergence of Gandalf the White is intended as a divine second coming of sorts - God's messenger of hope to aid Middle-earth in its hour of need. Gandalf not remembering his name in The Two Towers creates a distance between he and the other Fellowship members, enhancing the character's mysterious origins and signifying how much wiser and more powerful the wizard has become during his time dead. Gandalf's detachment from his former identity also ensures that dying carries a cost in The Lord of the Rings - even for a Maia.