This is not very well documented it seems.
Something like this seems to work though:
from arcgis.gis import GIS
gis = GIS("home")
import requests
olduser = gis.users.get(<username1>)
newuser = gis.users.get(<username2>)
with open("<filename>",'wb') as f:
f.write(olduser.get_thumbnail())
newuser.update(olduser.access, olduser.preferredView, olduser.description, olduser.tags, r"<filename>", culture=olduser.culture, region=olduser.region)
This is just the basic procedure of course, needs some more work, e.g. getting the correct file type from get_thumbnail_link(), and if possible cutting out the step to save the thumbnail to disk.