I don´t understand how monitor calibration works

To expand on Johan's answer, on windows very few native apps are color managed. You cannot trust the default browser, the photos app, etc to show you the right color. Only color managed apps will show the correct color. Those are things like Photoshop, Lightroom, and others. Firefox when the secret setting is enabled will color manage but internet explorer and Edge don't.

Lightroom is a color-managed application. It uses the monitor profile, period. There is no special way to confirm that because you do not need to confirm it.

Images will show correctly in every color-managed application. They may not show correctly in non-color managed applications, but that will also be the case if you do not calibrate and profile your monitor.

You are indeed getting it wrong. Color management does not work the way you described. But don't worry about it, Lightroom will automatically use the new monitor profile. There is nothing you need to do.