I think perhaps in this example "cuz" could be short for "cousin," although the comma seems misplaced. "Because" doesn't really make sense in this context.

"Cuz" is a slang word. Depending on context, it can mean cousin or because. It is also sometimes used between friends, indicating a close relationship similar to that between cousins. "Happy belated BIRTHDAY cuz, Anthony, hope to see you in a few weeks" Try to avoid writing it, if you must write it then only use it in text (SMS) messages.

The message lacks clarity! Hopefully the sender and recipient will understand it! Is Anthony a cousin? (cuz) Is the sender saying happy birthday (late!) because (cuz) they're meeting soon and he thinks he should say happy birthday? It really is not clear whether CUZ is 'because' or 'cousin.' Changes in punctuation, or extra detail, or CORRECT spelling ARE NEEDED here! This illustrates WHY PROPER SPELLING is important !

short form of because

Cuz is the same that Because In Spanish porque