I made it through multiple rounds of interviews but lost to another candidate. How do I process that without losing confidence?

It's a two-sided fit process. If you made it far into a loop, the problem wasn't your ability — someone else just had a closer match to what they were looking for at that moment. What you can't control is who else is in the pipeline. What you can control is your next interview. Let the rejection inform your coaching work, not your self-worth.