Depending on context and where you're from, it can also mean "get lost". I think that's how the OP was using it.I'm there, too--kiss off is what I'd heard pretty much all up to this thread. "Piss off" is not a "go away", it's a "made/got angry". So telling people to go get angry (from the op, substituting the meaning for the phrase) just makes no sense at all. The kiss off phrase gets stronger with "kiss off and die". So, two totally different phrases.When I was growing up, I often heard 'kiss off'. I didn't run into 'piss off' until I was old enough to be reading books set in England, and I still think of it as being more of an English thing to say. (ymmv) So it's not a matter of trying to be 'cutesy', it's a matter of having heard and used one idiom for a long time time before learning of the other. And no, I don't secretly mean 'piss', I mean 'kiss'. As in giving a big, sarcastic smackaroo in a person's direction as they depart.
I have no problem with people swearing. I have no problem with people not swearing. I don't swear myself. However, I dislike "cutesy" swearing. You know you mean p* when you say kiss off, I know you mean p* when you say kiss off, so why do it? Unless there was a child around, it seems pretty useless to me. We all know the feeling behind it.
If I have made someone angry, I have p*ssed them off. If I am angry myself at something or someone, then either I am p*ssed off, or will say that "X p*sses me off."
If I want someone to go away and I'm angry/vehement about it, I will tell them to "P*ss off!" Note that between friends this can be jocular, depending on the circumstances, but never so with strangers. It can also be a defensive reaction to a verbal challenge.
If I am p*ssed, I am drunk, not angry.
I don't normally use 'kiss off', nor do I know anyone who does much, but from the contexts I've seen it used (usually US usage) I'd expect it to mean something closer to brushing someone off, rather than telling them to go away. The construction might be something like "X kissed me off." to mean X dismissed all my concerns and comments and was obvious about not wanting to listen to me.
We have a wonderful language, don't we, with all these multifarous meanings?