How do you build confidence for interviewing at big tech companies?

Boring answer but really the main thing is to just get as much practice as possible in the kinds of questions they're going to ask you. It's good if you can get specific feedback so in that sense one of the online interview practice sites where they pair you with someone might be beneficial than just practicing a bunch of leetcode. As for getting past the anxiety you might feel on interview day realize that it is normal and have the mindset that you're going to approach the problem as more of a collaboration with your interviewer where you discuss your approach and write some code rather than just a strict evaluation. Most people really do want to help you get through it from what I've gathered so it pays to leverage them by asking questions and checking to see if they think you're on the right track.

In my experience it helps a lot to not think of it as an interview, but as a chat with just another coworker, and get genuinely engaged in what’s being asked.

I interviewed in most FAANG and got offers by just really engaging with the interviewers. Especially the system design rounds I found very fascinating and I treated the interviewers as if they would be my peers, challenging them (positively) when appropriate, and quickly admitting a mistake I just made.