Yes, they do mean the same thing. It means to keep something easily accessible. I would use 'keep it handy' more than 'keep it on hand'. I'm a more British flavoured native speaker, and I'm not just answering questions to tout for business.

Hello Davood. I would say "keep something on hand" or "keep it ready." I use "handy" as an adjective to describe a person who is good at fixing things or an item that is practical and useful. I would never say "Let's keep (something) handy." That sounds weird to me. I am a native English speaker who grew up in Oregon but has lived on the east coast and in the Midwest. I am also available for English lessons. Cheers!