No two guitars are the same.
Absolutely true! But that begs the question.
Broadly speaking, there are characteristics that certain types share. There are differences between these types in frequency response, resonant peaks, attack of the note, formants, and dozens of other factors that add up to the ability to categorize different types of instruments in certain ways.
I'm sure you'd agree that a Tele doesn't sound like a Strat, in spite of the similarity in their necks, assembly types, etc. The differences are there to hear due to the differences in hardware, body, pick guard vs body mount of the bridge pickup, the pickups themselves, etc.
Granted that two Teles will sound a bit different. But you won't be confusing any of those Teles for a Strat. or a Les Paul. The differences between types and models is greater than the differences between two of the same model.
A Vela doesn't sound like a CU24 or 594.
Dig a little deeper, and you can get to differences in more detail.
Humbuckers and single coils have characteristic sounds. Maple fretboards sound different from rosewood, which sounds different from ebony. Coil splits sound a little different from true single coils or P-90s. Guitars with a longer scale length are a little twangier in the bass. Trem-equipped guitars have a nice ring due to the springs. A solid body sounds different from a hollow body. A thick solid body sounds different from a thinner one. Potted pickups sound different from unpotted, and they're less microphonic. Ceramic pickup magnets sound different from alnico (and the same is true of ceramic magnets in speakers vs alnico).
After a while of owning or playing lots of guitars, characteristics of certain woods are pretty clear. My maple bodied acoustic doesn't sound quite like a rosewood acoustic, and that doesn't sound quite like a mahogany one.
Put those same woods on an electric guitar body and it's still possible to hear how they're different. My 1965 all mahogany SG Special with P-90s, sounds quite different from my 594 with P-90s and a maple top; the woods are different, the body of one is thicker, etc.
So I think it's fair to answer to the OP, yes, trem-equipped guitars sound a bit different from guitars with wrap tailpieces and explain those differences.