It was added recently, but it's easy to miss if you don't follow the release notes (here or on web). I'd also support having a toggle available directly while viewing the map, rather than only within the menu of a specific folder.

It would also be great if the icons were reflected in the display of saved places as well, while using a different marker color to distinguish it from search results.

I have a folder with places I'd like to visit (various categories), which I keep permanently visible in case I happen to be passing by. However, at first glance, the map just shows a bunch of stars.

After so much time of using the app it’s only something I see now. Would it be an idea to add a toggle to the “map layer” menu? the one in the bottom-left (on the app?)

Since you have Premium, you can enable permanent display on the map for individual folders, both on the website and in the app. However, this can't be enabled globally for all your saved places, and the points won't be distinguished by the category icon they belong to, as they are when you open My Mapy – they'll only be shown as a marker with a star icon.