How to get less space around text inside text box with rounded corners in InDesign?

InDesign can only place type according to the matrix of the glyph, which is a rectangle encompassing the ascenders and descenders, so when you have a rounded corner, the first character cannot fully appear on the first line until its top left gets past the edge of the frame (top example). You can try select a different baseline option between Cap Height, Ascending or x-height, and perhaps add a minimum to push it down so it can tuck closer to the left, at the expense of it being farther from the top edge.

The best may be using x-height as the baseline option (you'll still have to add a bit of minimum space to push it down a bit). In my second example, you can see the xheight (cyan line) now defines the top left starting point of the first character, and so it can get closer to the left. It's trial and error to get what satisfies you, and it's not without its problems: as you can see the top of the letter A is falling outside the frame now, so pick your poison!

Checking in: Experimenting with a thick stroke with rounded corners pasted inline below the text, and using baseline shift.

If the text frame is wide enough then you can make it as short as you want. But if it is so narrow the text comes close to the edge the text will be inside the curved sides and you will not be able to fit all the text in the frame. Also, since the text areae is not a rectangle you cannot use any Vertical Justification other than Top. You can stil use baseline shift or first line offset to centre the text.

Another tip: Set Vertical Justification to Centre and set First Baseline Offset to Cap Height.

Have you tried pasting a square textframe inside a rounded corner textframe?

In the frame's baseline setting try to change the first line start and chnge to cap's heights.

Did you check all he text frame options? (inset, baseline options)

Reduce all text spacing; reduce any text frame spacing. I don't think anything else controls that margin.