I believe I have a good relationship with a coworker. My manager thinks we don’t. Should I ask the coworker to record a formal feedback that we have a good relationship, or are there other standard or recommended data points or actions for making this case?
This is how you cure misalignment:
To make things clearer, here's some example scenarios after you talk with your manager:
Managers are extremely busy people. They miss stuff and get stuff wrong all the time. Not because they're lazy or mean or inattentive, but because being a good EM is insanely hard. So the onus falls on us as ICs to overcommunicate and keep them informed.
Here's 2 resources I recommend to help out here:
What about something in the middle where you informally say over slack, "Hey just wanted to confirm, do you also feel we have a positive work relationship?" ... "Ok thanks, mind if I share this exchange with my manager?"