Desculpar-se = to apologize
Desculpar alguém = to forgive someone
Except in standard greetings (I wish you a Merry Christmas), "wish" is associated with an unlikely event in the present/future or a past regret about something that happened or didn't happen.
She would like me to apologize. (a simple statement of fact)
She wishes that I would apologize [but I probably won't].
Thank you for your excellent answer. In general, I need to think about things that are unpredictable in the past, present, and future, in case those things happen or not happen, isn't it?
Since in English "say I'm sorry" and "apologize" are essentially interchangeable, why can't it be "She would like me to say I'm sorry." Other than the fact that my translation could probably also be said another way, the two English versions, Duolingo's and mine, are essentially interchangable, so why can't the translation also be "say I'm sorry?"