I socialize with people from every team. That's the way it is in our pool room, we have between 100 and 150 people playing league in any given session, and most of us know each other. Many of us have played on different teams together at one time or another. At the least, we see each other a couple times a week, year after year, so we know each other. New players tend to get introduced around to other folks, most of the time. Because we've all become friendly, having played league together for a long time.
If you were to be in a league for more than a few weeks, you would start to become known to other people playing. As to whether they would socialize with you or not, is up to you and how you handle yourself, I suppose. I've found people in general to want to be friendly, of course there are exceptions to that, like anything else.
You can't expect it to be like that for you after a week or two. If you play regularly, people will begin to know who you are. If you are pleasant to them, and make at least small conversation, over time you will probably get to know some of them. You can't force friendship; you need to be a friend (or at least friendly) before you can make a friend.
And yes, some people will spend their time at league on their phone, or at the bar watching the ball game, outside smoking, or any number of things. I look at my phone some thru the night. Perfectly natural in 2020.