Is it possible to catch Mewtwo more than once?

If you make Mewtwo feint he will disappear from the cave. The way to get him back, as far as I know, is to go and beat the Elite 4 again upon which he'll return to the cave.

If you capture him, can you go repeat the Elite 4 and capture a second Mewtwo from the cave similar to how accidentally beating him will spawn another after the Elite 4 are beaten?

This would be particularly convenient for trading the X or Y Specific mega evolutions with a friend or family member who has the opposite game.

Edit: I guess this question could also pertain to other legendaries like Zygarde, Moltres, Zapdos & Articuono.

Solution 1:

No, once you have caught a legendary (that is able to respawn if beaten), it won't respawn again in the same game.

I believe this done to prevent having dozens of legendaries from the same game (in the sense you mentioned, i.e. trading one away, get another, trade it yet away, get yet another...), while at the same time, give players the possibility to catch them if they accidentally faint the legendary Pokémon.

Legendary birds don't respawn in XY. Mewtwo and Zygarde can, however.