Is a chestnut and a buckeye the same?

No.

Buckeyes belong to the genus Aesculus (family Hippocastanaceae). Chestnuts belong to the genus Castanea(family Fagaceae, which also contains oaks and beeches).

You are probably confusing chestnuts with horsechestnuts (not related), which are also Aesculus, like the buckeyes. The family name, Hippocastanaceae means, "horsechestnut."