They are very similar but it does depend on the context. 'Testing my patience' is more formal. 'Annoying' is more casual.

They are not the same. If a cook puts too much salt in your dinner, he may annoy you but he won't be testing your patience.

Yes, they’re the same.

Hi Angela! The two sentences have a similar meaning, but "He/she is really testing my patience..." is a little more polite because it is less direct. I hope this helps!

They're both correct but would be used in different situations. If someone is annoying they are doing something that you find irritating, tedious or frustrating. It could be important or trivial. To be 'really testing your patience' they would be pushing you hard on something; this is stronger than being annoying.