What does it mean to be someone's season?

A season in this context is indeed a TV season.

Bachelor in Paradise has participants--young men and young women--from previous seasons of The Bachelor and The Bachelorette. The TV season in which they first appeared is identified by the name of The Bachelor or The Bachelorette who was the lead in that season, which is explained in the following paragraphs.

The Bachelor first appeared on American television (ABC TV) 24 years ago. This is reality TV. In each season, someone is selected (by the producers of the program) to be The Bachelor. The producers also select 12 (I think that's the right number) young women. The program airs weekly; each week The Bachelor rejects one of the women. The one remaining is his first choice, and the two are expected to become engaged to be married. A season is, as I recall, 13 weeks.

The Bachelorette, which has not been on for as many seasons, works the same way: a young woman is The Bachelorette, and she interacts with and chooses among a dozen or so young men; one of them is the winner. (The rejected person each week fails to get a rose.) Each season is filmed in a different, often exotic, location, e.g., Brazil, Paris, etc.

So there are, from each season of the two programs, a dozen or so young men and young women. Some of them return to be part of a later season, or to be the lead--The Bachelor or the Bachelorette.

The seasons are identified as, e.g., "Hannah B's season", because Hannah B. was The Bachelorette, and the young men from that season are from Hannah B's season. Colton's season is the season when Colton was The Bachelor.

The Bachelor in Paradise (commonly known as BIP) has participants from various seasons--from Connor's season, Hannah B's season, or whenever. I understand that people will also be rejected as the season progresses, although I don't know what the process is on this one. From a quick look, I gather that there are three more women than men, and so three of the women have to be kicked off soon. You may know more about the BIP plot than I do.

There is a great deal of chatter on the internet--Twitter, Instagram, etc.--on what's happening, who's dating who, violations by participants of the rules, etc.

I personally have never watched an entire episode of any of these three programs, but they have been extremely successful and involve a lot of attractive young people getting acquainted and dealing with romantic relationships.

You can search the key words on Google, e.g., https://en.wikipedia.org/wiki/The_Bachelor_(American_TV_series)