There are also countless hours of opt-in training for pretty much any subject where you want to improve your skills.
In Google SRE, combo TLMs are considered to be an acceptable short term solution for team turnover, but not a long term best practice. TLMs are highly encouraged to find a different TL for the team.
In addition, all SRE managers (SRMs) are paired with an experienced manager as their mentor.
Ultimately though, apart from the mandatory trainings, no one can really force you to be a better manager. The big feedback mechanism is that internal mobility is very high, to the point where managers have essentially no power to prevent anyone from leaving their team. So if you suck, you will get bad scores in your own performance review and everyone on your team will just transfer away.
I called out the SRE-specific bits, and a few common practices, but it could be that there are different practices in other engineering orgs.
Source: I'm an SRM, and speak only for myself.
Also, right now, I believe the norm is to have managers and TLs as separate positions, even in small teams (like mine, 5 people in total).
I just went through a 2 day required training as a new manager. This is required training that has to be done within the first 6 months of you becoming a manager.
This has been a requirement for many years.
Source: I'm a googler
Edit: just to add, the same training is required for TLMs. The moment you get a direct report the training becomes mandatory. I suspect there must be some miscommunication from the folks you spoke with.
The way it's been described to me is that most managers at Google are TLMs - Tech Lead Managers. They are responsible primarily for the tech lead part and are only coincidentally responsible for the management part. Some teams do have separate Tech Leads and Managers, but for most teams, it's a single person.