I installed the filter made for dodge caravan on my wife's corolla. Both filters are similar in size (the caravan's filter is a little bigger than the one for the corolla) and the threads match (that is why I did not notice the mistake).
Can the mix-up cause any damage to the engine? Thanks.

No idea, but I would replace it with the correct filter and add some oil ASAP.

It's possible as the relief valves might be different but why chance it for the few bucks a filter costs?