As with almost every game on the switch, your save data is tied to your user account. Each user profile on the Switch has their own save data.

However in addition to this, with the new Switch Online program, Smash Ultimate supports Cloud Saves which will backup your data on the Nintendo account your switch account is linked with. This means that you’ll be able to pull save data down from the cloud onto another switch if you so desired.

So your save data is both linked to the local User Profile on the Switch and (if you use Cloud Saves) to the Nintendo account the User Profile is linked with.

Update: As of update 7.0.0, Garreg Mach Monestary is different

For whatever reason, the stage itself is a positioned one unit higher on the screen. So the top blast zone is one unit closer and the bottom blast zone is one unit further.

This is thought to be a bug that will end up patched out, but until then, the stage's Battlefield/Omega forms are banned in tournaments.

You can see more on this tweet: https://twitter.com/Lyra_GP/status/1223989295830179842

Below you'll find the old answer:

No, all Battlefield/Omega stages have exactly the same bounding boxes and dimensions.

From the Wiki Page:

In the introduction of Ω form and Battlefield form stages in the E3 2018 Nintendo Direct, Masahiro Sakurai stated that all Battlefield forms will be identical, differing only in music and aesthetics.

(note that all Battlefield/Omega stages will be 50% larger when played with 5+ players as in Smash 4)

However, there is one difference that affects gameplay

Smash is actually a 3D game, meaning the hitboxes and hurtboxes of moves and characters all have a Z-axis attached to them. This causes some weird interactions, like this one:

But some stages, like Flat Zone, are "2D" and have no Z-axis. To represent this in the game, the stage has a "flattening effect" on all hitboxes and hurtboxes that pulls them all into the same point on the Z-axis. This 2D property persists when you choose the Battlefield/Omega version of a stage.

For a list of stages with this property, see this reddit thread: All Stages with the 2D Effect