Just tested this, this happens to me too, and it’s not just a performance issue. In my case, the second tileset had the animation faster and the faster animation overrode the slower one. This seems like a bug, perhaps Tiled is using the images as tile identifiers rather than doing it properly like it does for regular tilesets…
Fortunately, there is a workaround, and it’s actually the intended way to do what you’re doing:
Instead of using multiple collections, assign your alternate animation to a tile that was another frame of the original animation. The animation can still start with your intended frame 1, the tile the animation is attached to doesn’t have to be the first frame. This way you’ll also have an easier time telling apart the animations you’re using since they won’t be in identical-looking tilesets, but will be different tiles in a single tileset. There’s no reason to have these duplicate tilesets just for slightly different animations :]