By the way, no thrush so far. My boyfriend has subsequently developed esophogeal candidiasis. We're getting tested but can't get an appointment for two weeks and won't get results for another 3 weeks. A white furry coated tongue does not sound like thrush. Thrush is not contagious.
Esophageal candidiasis is a sign of severe immunodeficiency.
If your boyfriend has never been tested, then I'm not sure how it is that you know he had that. I can't imagine that a doctor would have made such a diagnosis without performing an HIV test. It sounds like you're making a lot of assumptions and guesses. But it wouldn't be a bad idea to get tested. Experts appearing on this page are independent and are solely responsible for editing and fact-checking their material.