is there a service where you can send requests from any website and it will store all the data and create graphs?

E.g. send a request when somebody press 'download' or 'buy' button so in the end you could see the chart of such clicks.

Yes. I think sites like mixpanel.com/ are built to do that

I heard that google tag manager is doing that, I am going to give a try to google tag manager

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