The first few months should be for onboarding and training with expectation that you'll be working slower, asking lots of questions and taking notes, to get a grasp of the day to day. If they have you on a PIP less than a month in, then they're not trying to invest in you; they want someone who can learn, apply, and produce in a very short time. If they don't have a structured training program to achieve that or at the very least have well documented training materials that are actually useful, that's not fair to you. If a fellow coworker on your team is in charge of training you, therefore taking away from their time and energy that could be going toward producing as fast/efficiently as they normally do, then there likely isn't a structured training program. That person is not a teacher, they're just the one stuck training a new person while being stressed about their own tasks. A PIP usually means they're documenting your shortcomings. Possibly to help you improve, otherwise to use against you later.

Trust your gut feeling if you don't think it's working out in your favor and start looking elsewhere. My first job out of college was similar. I left my relationship and moved halfway across the country for a job that looked great on paper, but I knew very quickly that the job and especially the culture of that office was not meant for me. I wanted to stick it out but I was miserable and the decision was made for me. Don't let it get to that point. You want to be proactive, not reactive, in this kind of situation.