It is a cute name, but I doubt most people will use the full name anyway, so it would probably get shortened to just Elliot or Jack. Or people may call him by his initials, EJ.
In some areas, like the south, it's common for people to be given two first names and go by both, not just one. Agreed though that sometimes other people are inconsiderate and call you something that isn't your name or is a nickname you didn't ask for
Re: How do y’all feel about the name Elliott-Jack?