Can is used informally to ask for permission. May is used for formal conversation to ask for permission to do or have something. for istance, a common phrase is "can i go to the bathroom?" That wording is informally asking for permission, but is not reliant on whether or not the person has the ability to perform the action. for the question to be GRAMMATICALLY correct, it would be worded "may i go the bathroom?" but it is used in everyday conversation as Can. hope that helps!

For all the examples I could think right now it seems to me that "may" sounds more like asking for permission or to be polite like "may I ask you something?" or "may I go to the bathroom?"

In ordinary spoken U.S. English, there is no difference, and it is more common to use "can." When I was in school in the 1960s, I was taught that "can" implies possibility, "may" implies permission. For example, suppose there is an unlocked door with a sign on it that says "Authorized Personnel Only." (Assume you are not authorized). Since it is unlocked, you can enter, but since you are not authorized, you may not enter. In real life few people pay attention to this distinction, and it may even be a "grammar myth." I see that the American Heritage Dictionary has a long and helpful usage note about this: https://ahdictionary.com/word/search.html?q=can

"The only difference between the two verbs is that one is more polite than the other. In informal contexts it’s perfectly acceptable to use can; in formal situations it would be better to use may."

Oh, boy this is a question we tease young English kids with May I have another biscuit? Yes here it is. Can I have another biscuit? Yes you can but only when you ask nicely.