purpose of egg white in gin fizz?

Try a Savoy Hotel White Lady cocktail - use vodka instead of gin. Google it.

Emulsion, no. Foam, yes. Mouthfeel? Subjective.

Egg white is essentially protein. When shaken with the other liquids in the drink, the protein molecules stretch and capture air (like when whipping up a meringue) as the mixture emulsifies, giving texture and frothiness to the drink that is both visually appealing and has good mouthfeel.