I just bought the game when it was on sale and played a bit. Today, I boot it up only for it to ask me for my nickname as though this is my first time playing. What happened?

Only thing I can note is that when it exits to DOS I just alt-Tab and close it, not type in "exit." Does that have anything to do with it?

The nickname is your in-game profile of sorts. Just type in the exact same name you used before and you will find your saves. If you write a new name you are creating another profile with a unique set of save game slots.

Having to close DOSbox manually is an inconvenience for everyone, but it is not affecting the game itself.