Also worth noting is that the cheese product sold as 'swiss cheese' in the US has a very indirect relationship with Swiss cheese, i.e. cheese from Switzerland. It's supposed to be reminiscent of Emmentaler, but the differences are greater than the similarities. (At least in my rather limited experience.)
There are different cheese mixtures used when making fondue, and while some people use Emmentaler, it's by no means universal. (I'd go so far as to say that it's a minority ingredient in most fondues.) Personally I'm rather partial to 'moitié-moitié', which is a mixture of equal parts Gruyère and Vacherin¹.
But that's a question of personal taste, rather than cooking, and as such is border-line off-topic here.
Of more interest is what happens in cheese fondue, and what kinds of cheese can be used. Hervé This looks at this in his excellent book Molecular Gastronomy:
A fondue is no more than cheese heated with wine. The combination of
water (from the wine) and water-insoluble fat (from the cheese) means that
the successful fondue is necessarily an emulsion, a dispersion of microscopic
droplets of fat in water solution
Connoisseurs of fondue know that the success of the dish has to do particularly with proper cheese selection. Questions of flavor come into play as well,
but well-ripened cheeses are best suited to the preparation of fondues because,
in the course of aging, enzymes called peptidases have broken up the casein
and the other proteins into small fragments that are more readily dispersed
in the water solution. These casein fragments then emulsify the fatty droplets
and increase the viscosity of the aqueous phase
... select very dry wines—indeed, wines that are excessively acidic and, if possible, very fruity. Why are these properties useful? Athony Blake has shown
that such wines have high concentrations of tartaric, malic, and citric acids.
Malate, tartrate, and especially citrate ions are very good at chelating (or sequestering) calcium ions. The acidic and fruity wines experts prefer help separate the casein micelles and release their constituent proteins, which stabilize
the emulsion by coating the fatty droplets.
(Another option is to cheat. Add some cornstarch, and it's highly unlikely to seize.)
¹ That should be Fribourg Vacherin, rather than the Mont d'Or variety.