Does " since forever" make sense?

It is an idiom. It is correct and is used to mean "since a long time ago".

Forever denotes an endless period of time that could be in the past, the future, or from the past to the future. Putting 'since' in front of it makes it be just about the past. It literally means 'from the beginning of time until now'.

I think you got a good answer. I will add that I think it is a term of slang hyperbole. It is an exaggeration common with teenagers mostly. As teenagers are experiencing time and words and the the whole universe in new and developing ways, they speak in overstated expression. But anyone can say it that wants to suggest something has been dragging on and they feel there is no resolution in sight.