Resolve liquid variable leakage between components

For now, this issue is mostly for myself (so I don't forget)! I'll flesh it out in a bit :) 

Context: https://github.com/just-the-docs/just-the-docs/pull/1058#pullrequestreview-1229786497 and https://github.com/just-the-docs/just-the-docs/pull/1058#issuecomment-1374268461.