The line between tech lead and engineering manager can be a bit blurry, depending on your organization. An EM could do a lot of the things you describe… but EMs should focus on people management primarily.
How does the org chart look? If the CEO asked HR, is your team directly reporting to you? Do you approve time-off requests? Do you know what your budget is for salary/bonuses/team lunch/etc?
You wrote about 1:1’s… What are you talking about? Are you trying to figure out that person’s career goals and helping them get where they want to go? Are you helping them figure out their next promotion?
Are you writing performance reviews for your team? Could you hire or fire someone if called for?
Those are a few EM behaviors I can think of off the top of my head. A lot will depend on the size of the company too. If you are at a 50-person startup then not much people-developmental activity is likely to be taking place. And if you are at a Fortune 500 then you would not be asking this question :)
This answers the question then. You are not a manager if nobody reports to you. Tech Lead it is.
It sounds like you are filling a void on your team and doing a good job at it. Keep at it!
I’m going to kindly disagree with other assertions on this thread that titles are bullshit. There are definitely bullshit titles (like practically everyone at a bank being an Associate VP).
A title that reflects your level of responsibility is important when you move to your NEXT position. That is how you mark progression in your career. It saves you a long conversation explaining how your title was (for example) “Senior Software Engineer” but you were acting as Tech Lead or Architect.