If I submit an answer only to find out that someone just answered X seconds before me with the same answer, should I delete my answer?
If you add something slightly different or approach it differently, I'd leave it. If it's exactly the same - I'd lean towards yes.
This might happen when you are composing your answer, especially for common topics and technologies, after posting it you discover that only few moments ago, someone else answered it, same to what you just answered.
In this case, for me personally, I will delete my answer, Upvote that answer, and then I might add a comment if there is a small tiny detail I wish to add to that answer just out of courtesy
Actualy I would do the opposite: keep your answer.
Two answers are never exactly the same. even if you didn't add any detail, it would be better and clearer for someone to read both answers. Let the voting system decide which one is better.