“Are you seeing anyone?”
You're having coffee with a good friend, who you haven't seen in a long time. You want to ask if he has been going on dates or been romantically involved with anyone.
Are you seeing anyone?
"Seeing" someone means dating them or in a romantic relationship with them. For example:
Are they seeing each other?
I was interested in her at the time, but she was already seeing someone.
You might not use "seeing ___" when talking about the past:
That was back when they were dating.
You also wouldn't use "seeing ___" to talk about a couple who has been together for a long time:
They've been together for, what, like three years now?