How to replace the remote device for a garage door opener?

I just spent the past year renovating and rebuilding my Mom’s house. There’s one, final item remaining – the garage has a functioning door opener, but its remote, hand-held operating device is long gone.

Are there universal objects to replace the original? Or, do I need to identify the specific make/model and contact the manufacturer? Will I have to “program” the device so that it won’t open every garage door (or, so that every other device won’t open this garage)?