I have not been able to find images that matches the Griffen on your canns/mugs. The milled decoration was very popular during the early 19th century and the method of construction matches the period. I will continue my research and will get back with whatever information I can.
If the maker's mark on your mugs is exactly and only "GRIFFEN" then it is problematic whether any of the makers I have cited made your mugs. I too have been unable to find reference to a maker that used the mark "GRIFFEN".