Depends on the team — managing can be quite a bit more scope than being a senior IC, depending on expectations for that role. You have broader ownership of technical outcomes over time, even aside from the extra responsibility for growing a team. Managers have all the responsibility of a senior engineer plus more. In that way manager feels to me like a clear promotion to me. Manager vs staff eng, maybe not though.

Management not being a promotion doesn’t mean that managers aren’t (usually—I’ve both been at equal and higher levels than my managers at times) higher levels than their reports. It means that switching to a management role from IC is never a promotion itself (ie always L6 -> M1 in Google/Meta levels) and it never comes with any difference in compensation.