Jonathan Scholbach explains Solidarhafter.
Mitbewerber is any co-signer of a lease who is not married to, not in a civil union with (eingetragener Parter in your list), or a child of the lessee. For example, if a non-married couple or two friends want to rent a flat and both want to sign the lease contract, both are Mitbewerber of the other.
Why is there no option for "family/relative"?
Family in the legal context of renting an appartment are the spouse and the children of the tenant. These are both mentioned in your list ("Ehepartner" and "Kind"). Other relatives, such as siblings, cousins, or uncles, are not family members and are considered Mitbewerber like non-related persons.