logging: refactor: pull prefix code out

Summary:
This is a partial backport of core#28318
https://github.com/bitcoin/bitcoin/pull/28318/commits/c5c76dc615677d226c9f6b3f2b66d833315d40da

Depends on D18268

Test Plan: ninja all check-all

Reviewers: #bitcoin_abc, Fabien

Reviewed By: #bitcoin_abc, Fabien

Differential Revision: https://reviews.bitcoinabc.org/D18269