Mechanically, A is simply unable to see B. Narratively, the players and DM need to figure something out that doesn't sound absurd or break immersion. Perhaps B is in the rafters, hiding inside shadows, or is able to temporarily distract A.
I agree that mechanically it sounds really frustrating to think that people are literally turning invisible, but think of it more like Batman or Spider-Man sneaking up on someone, even from the front. They will figure out some way to get past A because of their incredible skill at Stealth. Mechanically, it is so much simpler to say that Batman was Invisible rather than having a whole separate condition.
I would just shrug and say that a person who cast See Invisibility or True Seeing happens to be using Divination magic to notice non-visual queues which happens to include stealthy people. Anyone can detect a sneaking person by succeeding on a Search action, but it happens automatically for creatures with special senses.
I personally would add a house rule where creatures are constantly doing a no-action-required Search using their Passive Perception score simply to allow for creatures like dogs with enhanced Perception to smell to automatically detect invisible creatures. I would hope that something like that would be included in the Dungeon Master's Guide later. If a PC with a high Passive Perception encounters an Invisible creature, I would hope the DM would either just tell them directly or at least tell them to make a no-action-required Perception check.