Seeing someone basically means the same thing as dating, but there is a small difference between the two terms. Seeing someone suggests that the relationship is not as advanced as when you are dating someone. Simply put, you are seeing someone to decide whether you want to start dating them.

Hans and Conrad have given you the correct answer. As to your follow-up question, the answer is no. In this context, "seeing" is an action verb that means a less serious form of dating. It's not a mental state.

It means you are dating someone on a regular basis but your relationship is not official yet.