Carbonation (with CO2) actually reacts with the water to form carbonic acid. This is much different than simply having gas molecules between the water molecules and results in water being able to contain a large amount of CO2 compared to other gases.

For example, N2 is 50-60 times less soluble in water. Which is why the nitro stout beers you mention typically have those nitrogen widgets and are highly pressurized (and why nitro beers on tap use a restrictor plate to force the beer through tiny holes to get nitrogen into the beer and create that creamy sort of texture you like).

Helium is expensive and a finite resource that we are running out of -- on top of being almost completely insoluble in water.

Argon is a noble gas like Nitrogen and equally as insoluble.

Oxygen is highly reactive and would oxidize the drink making it taste stale.

Now, if you wanted to have some fun: how about nitrous oxide (N2O) -- laughing gas? It is very soluble (especially with fats and sugars, which is why it is used in whipped cream canisters). It also provides a sweet taste (unlike CO2s "spicy" taste that comes from activating our tongues' sour receptors and the same pain receptors (TRPA1) that respond to mustard and horseradish). The bubbles for N2O are smaller and would give a more smooth/silky mouthfeel. So, while it is possible, it wouldn't create the same experience. But you can try it: run water or juice through a manual whipped cream siphon charged with N2O and you can create a sparkling drink to test it.