Yes, both players have the advantage of using the same clock. However there is an argument that the player that does not own the clock might be so unfamiliar with that clock that he has a disadvantage. You could and should say any advantage of clock familiarity is minimal at most. Well the same is the advantage, during the game, of using an electronic score keeping device if it is used for score keeping only, as it should be.
Hmmm…I wonder if one could try to arrange so it DOES confer an advantage?
First, get a truly hideous chess set. Something like this Super Mario set https://www.chessusa.com/product/44-033.html, or the horrid Peter Ganine “Gothic” set http://chessantiques.com/product/sculptured-gothic-chess-set-ganine/.
Then, bring it to your tournaments and try to use it. It’s non-standard so if your opponent has a more standard set they can force you to use their set, but sometimes there won’t be a standard set available and yours will be used.
In those games, I can see having a 2D board to look at could actually confer an advantage.
Or simply becoming used to the horrible set would confer an even greater advantage without the need for an additional device.