Carol's questions to Fury were probably more testing how "realistic" they sounded and his response rate and confidence. People can usually easily recall the details of the questions she asks him because they are about major events in their life. Therefore, answering them would be quick like Fury does when replying. However, a Skrull can only access recent memories and so would likely struggle on the spot in the quick fire rate the questions are coming in to come up with answers that sound believable.
Carol: Skrulls can only assume recent memories of their host bodies.
Note that there is also a scene shift afterwards with a time skip and so Carol might not have trusted him 100% in the moment but fact checked somehow later on. The fact he gave actual real place names and events would help with that.
For proving Carol is Carol she shoots the photon blast and whilst this wouldn't make Fury 100% sure she isn't a Skrull because he doesn't know them, it should help. Remember when they first meet a Skrull shoots at the pair of them but doesn't use photon blasts. If Skrulls could do that they wouldn't carry weapons, which is part of the reason Carol doesn't. Why carry bulkiness and weight when you yourself can do much better? So he should be somewhat confident she isn't a Skrull from the demonstration even if he isn't fully bought over.
On top of that it could also be noted that Fury is now in no position to fight back against her, 9 times out of 10 she is going to beat him so safer for him to play a long game if he doesn't trust her. From the same perspective Carol could have done it to intimidate Fury to try and get him to do what she wants without having to reveal any information.
It's also worth noting that this test would never be 100% accurate and is more of a quick test to see if they can trust each other. It isn't foolproof but more of a gut feeling type of test. And for what it's worth I got the impression they were both somewhat sure each other wasn't a Skrull at the start of the scene but did it as more of a formality and to get to know each other a bit. It appeared to be more of a trust test than a species test.