Generally, is daily commit and push necessary for managers and
supervisors to check my progress and productivity? Is it especially
true in Agile?
Well, it's good practice to commit code at least once a day, to ensure you're not keeping hours of valuable work hostage on your workstation!
But, for "true agile?" No.
For managers to "check your progress?" Not really. A better check of progress and work ethic is velocity over time and 1-to-1 dialogues.
That said, we don't know your team dynamic. And, one of the most basic principles of "Agile" development is, "do what works for you." Your team needs to discuss these policies internally (with your manager) to determine why or if this procedure is helpful.
And from my own experience as both a developer and a manager, regular checkins aren't necessarily a product of an overbearing manager. They can also be a sign of a manager (or team) who's actually interested in your work or wellbeing. Or a manager who's interested in not losing days of your work if you get his by a bus ...
I should also note, from my experience on both ends, that when someone — myself included — isn't making regular pushes, they've usually "checked out." That is, they're not actually working. And more importantly, they've stopped caring.
So, as long as you're actually doing work, and as long as writing code is a daily responsibility of yours, you shouldn't be afraid to push code every day. If you find that difficult, you might be in the wrong job! And, that's something both your manager and you should want to see the early signs of!
And I mean that seriously. When I'm in development mode on a project, if I get to the end of the day and can't push my code, it's never because my code doesn't compile or I'm embarrassed. It's because I didn't work. ... So, I either walk up to someone at the office and make a commitment to them1, which a great motivator, or I start looking for another job. (Or both.)
1. That is, I walk up to a stakeholder or my manager and say, "Hey, I'd like you to review my work at the end of the day tomorrow." And to be perfectly honest, after a few days of this routine, I usually stop looking for another job, because I'm engaged again. I care again, because I'm no longer working for a vague corporate entity. I'm serving my coworkers.