There are many fanfictions on this topic. In some, Tony Stark might be watching Peter Parker's growth as Spider - Man during the events of 'Homecoming' with a sense of pride, like a mentor observing his protege. He could be making mental notes about how Peter handles different situations.
Well, it all started when Tony was kidnapped and forced to build weapons. He then built a suit to escape and decided to use his tech for good, becoming Iron Man.
Yes, Tony Stark does know who Spider-Man is in the comics. They have interacted and formed significant relationships over the course of various storylines.
Pepper Potts could be a big help. She knows Tony so well, and she could try to jog his memory by showing him things from their past together, like pictures or mementos. Rhodey could also be there for him. He could tell Tony stories about their military days and how they became friends, which might trigger some memories.
One common character paired with him is Bruce Banner. They could be gaming buddies, with Bruce using his scientific knowledge to help Tony improve his games or vice versa. Another is Pepper Potts. She might be shown as someone who tries to get Tony to take a break from his intense gaming sessions.