Unfortunately, that is not possible directly right now. However, you can do it manually. For that you would regenerate the mesh with `ToElementMesh` and specify a BoundaryMarkerFunction and a PointMarkerFunction. So, the None needs to be replaced with a function that does what you want. (See the documentation of ToElementMesh in the options section to see how those are written)

    m2 = ToElementMesh["Coordinates" -> mesh["Coordinates"], 
      "MeshElements" -> mesh["MeshElements"], 
      "BoundaryMarkerFunction" -> None, "PointMarkerFunction" -> None]

**Update:**
The direct setting of boundary and point markers will possible in version 13.3 (not in 13.2).