'Polish' and 'polite' share a root — Latin 'polire' (to smooth). Polished surfaces and polished manners are one.
To make the surface of something smooth and shiny by rubbing; to refine or improve something.
From Old French 'poliss-' (stem of 'polir,' to polish, to smooth, to make fine), from Latin 'polīre' (to smooth, to polish, to refine, to adorn), possibly from PIE *pel- (dust, flour, fine particles), the connection being that ancient polishing involved rubbing surfaces with fine dust, powder, or ash to achieve smoothness. Latin 'polīre' also gave English 'polite' — literally 'polished' in manners, a person whose rough social edges have been smoothed away — and 'interpolate' (to polish between, to alter by inserting). The semantic parallel between physical smoothing and social refinement appears independently across languages: French 'poli' means