Along with the existing Panama Canal, what would have happened if it was joined by 2 other Canals in Nicaragua (similar to OTL proposal) and in Mexico via the Isthmus of Tehuantepec (the latter possibly as a result of some Dictator or some other scenario where Mexico fares even worse against the US)?
Well might find out in 10 to 15 years what the effects of a Nicaragua canal will have on Panama.
But from what I understand the Panama Canal is actually two parallel canals (with a third not normally used), I would say that depending on when each canal was built you'd have enough traffic for two canals. The Mexican one though was always planned realistically to be a railroad, involving constant unloading and loading and can't see that as efficient. The Nicaragua Canal if done around the same time as the Panama could force the Panama to finish the way the French began- as a level canal without locks; this is extremely time consuming and labor intensive and takes a lot more money, but would allow ships to be as long as you want them allowing the Panamamax to be bigger than a "Nicaraguamax".
But this is all moot... once the US picks one, they aren't going to allow a second anytime soon, local unfriendly Dictator or not, in fact ESPECIALLY if it is an unfriendly Dictator. Once the US has their own canal, if European country tries to then they will claim Monroe Doctrine as raison d'etre for stopping it. Only path to multiple canals is US seeing that it is in their best interest and they have control.