Honestly, it's up to you. There is no "right" answer here, although there might be a "right" answer for the company (i.e., they need you to step into a leadership position or else they need to find someone to be your new lead).
It seems like you're confusing "tech lead" with "engineering manager," because most tech leads spend a significant amount of time coding. The tech lead on a project is the person who shapes the technical direction of the project, knows all the ins and outs, and helps other people on their team from a technical perspective.
At many companies they might have 1-2 reports, often junior people who need active mentorship, but most of their time is spent on technical + project management work.
I recommend the book _Project Management for the Unofficial Project Manager_: https://www.amazon.com/Project-Management-Unofficial-Manager...
A very coarse generalization is that tech leads own the "how", engineering managers own the "who", product managers own the "what", and project managers own the "when".
At smaller companies these responsibilities naturally blur together. It also depends on the nature of the business. If the company has a big client services component with lots of custom integrations, then project management will almost certainly be its own dedicated function.
Give it a shot and see how you like it! Speak with your boss and say you're unsure of whether this is for you, but you'll give it a college try. The worst that happens is you learn you don't enjoy that type of work.