When ships 'move a class' it could be they received an upgrade, albeit being already close to the previous/next class, putting it in the previous/next class.
A heavy Medium Cruiser could after upgrading weapon system become a light Heavy Cruiser, but still a Heavy Cruiser per set definition.
The name: the Enterprise linage kept it's name, for 2 reasons: each Enterprise came from the previous drawings, the previous Enterprise.
The 1701-B is based upon the 1701-A, the 1701-C on the 1701-B, and so on.
The A series was probably so good, the Federation figured to reuse it's base, where other ship types had likely a more intrusive overhaul.
Considering the story lines: the Enterprise WAS a bloody decent ship, a survivor (albeit this had a LOT more to do with the crews than the ship itself: even a super cruiser woud bite the dust vs a fighter if that super cruiser had a bunch of morons on board, right?).
As said by Mr. King: the Enterprise is a tribute to past Enterprise ships.
Why the name and the serial numbers are like humongous?
Well, a lot of Federation ship captains could be still blind idiots, maybe it's just for them, making sure (hopefully) they are not targeted by 'friendly foes' (read: friendly fire)?
As for the Defiants: Sir, comparing the previous Defiants with the NCC-74205 is like comparing apples with Abram tanks.
There simply is no compare, but two: both fly in outer space and have a heap of folks aboard. XD