Anyway back to a Fleet Museum:
It comes from the episode Relics at the earliest (following is from memory):
Picard looks around the Enterprise 1701 Bridge: Hm. Constitution Class.
Scotty: Ay, you know her?
Picard: There's one like her at the fleet museum.
Which apparently now is the 'New Jersey'. Terrible name for it, but whatever, it does expand the verse and fleet a bit - not everything was done by the 1701s!
And it hearkens back to the 'Starfleet is totally not the USN/Navy of the UFP' thing since the beginning. While the USN doesn't have a singular fleet museum, Museum ships exist aplenty, and its just something we do: pride in our history, our institution, our achievements, to educate the future generations on what was achieved and what could be done. Many institutions, military or not, do this, have little museums.
Why wouldn't Starfleet have one? They're all about exploring, science, new phenomenea, human/alien achievement and diplomacy, having a Museum would be one of the least surprising things ever.