The Manhattan & Martini both derive lineage from the Old Fashioned. The Old Fashioned was so named in the late 19th century as it was trying to recreate the original cocktail, as in "Don't give me one of those drinks with all the stuff in it, give me an old fashioned cocktail".
The Martini derives from the Martinez which is a much sweeter drink. Heavy on sweet vermouth and using a heavier & sweeter style of gin. It should be clear at that point that the Martinez and Manhattan aren't super far apart.
The Manhattan & Martini both derive lineage from the Old Fashioned. The Old Fashioned was so named in the late 19th century as it was trying to recreate the original cocktail, as in "Don't give me one of those drinks with all the stuff in it, give me an old fashioned cocktail".
The Martini derives from the Martinez which is a much sweeter drink. Heavy on sweet vermouth and using a heavier & sweeter style of gin. It should be clear at that point that the Martinez and Manhattan aren't super far apart.