Just give the brush a name and tab off the name field and it will saved to your user profile.
But that is just one type of brush. It is a .vbr (parametric) brush. This is a vector so it is scalable without loss of quality.
There are more brush types. All of these are bitmaps:
1a) .gbr grayscale. White will be transparent, black will take the foreground colour.
1b) .gbr colour. Colours don’t change. Transparent is transparent, blue is blue.
2a) .gih grayscale. Multi layer animated brush. White will be transparent, black will take the foreground colour.
2b) .gih colour. Multi layer animated brush. Colours don’t change. Transparent is transparent, blue is blue.
3) .myp is a MyPaint brush.
4) .abr brush. This is a photoshop brush. Many, but not all, .abr brushes work in Gimp.
See
https://docs.gimp.org/2.10/en/gimp-conc ... ushes.htmlUse arrow top right to see a few more pages.