But it's not a 'completely different ship' though is it? It's the same ship that's been upgraded.
Comparing the 2240s Enterprise with the 2360s Enterprise-D is a false equivalency - these are two distinct vessels of different classes seperated by over a hundred years.
The Discovery is a ship that's had a significant refit, much like the Motion Picture Enterprise had a significant refit but wasn't a brand new spacecraft.
Except the Spore Drive (and general look of the ship), I mean, realistically (at least from the point of Trek) everything else on it WOULD have to be brand new - the infrastructure would be replaced with 32nd century materials, every single system would be effectively replaced with new state of the art stuff.
Think of it like programmable matter literally EATING the old and putting the new (which can already technically be done with transporters and molecular fabricators in 23rd century, and transporters and replicators in the 24th century).
SF overall updated the whole design as well without affecting the total look of the ship going to the extreme to be completely unrecognizable. A few support struts/connecting corridors were removed from the inner saucer section, a more galaxy class neck was added, etc. but internally, I don't see why the overall frame wouldn't be replaced as well.
So, technically speaking, it WOULD be a brand new ship - I mean, the technology could easily do it with ridiculous speed and efficiency that it wouldn't really matter.
Looking at the turbolifts for example, the vast chasm would be enlarged with TARDIS tech (if it was incorporated) and the rails for the turbolifts would be removed (they were maglev to begin with and at the time required tracks, but with 32nd century technology, you don't need tracks at all because the maglev/antigrav checkpoints materialize with programmable matter as needed)... so you gain internal space from removing the turbolift tracks, and expand on it further with TARDIS tech (although I'm pretty sure the writers forgot about the TARDIS tech and made an error, or the SFX department made an error in general because the vast chasm exists in seasons 1 and 2).