Angela isn't possessive— she just... Thinks that her healing is the best on the team and you of course deserve the best of the best.

She knows that there are other healers on Overwatch- it's just she has alot of experience! So when Brigitte tries bandaging your wounds Angela has to push the brunette aside and do it herself. It's not like Brigitte is incompetent- no!- it's just that after that mission she deserves some rest so Angela can just finish the job no problem.

"I'm glad your Injury was not too fatal, it would be a shame if they had to bench you." The blonde smiles as she spreads ointment on a scratch that made it's way to your forearm.

It wouldn't be a shame, really, it just means that she can check up on you more.