Generally, it is not possible to claim ownership of a frequency or band. However, repeaters are one very narrow exception to this. The owner of a repeater can say who is and is not allowed to use the repeater. This is just a guess on my part, but I assume this came about back when repeaters were much more complicated to set up and run. (E.g. there were repeaters that operated only on battery, or were owned by an individual to talk with his friends and family in a mountainous region, or had weird technical limitations.)
Repeaters which are open for use by anyone with a license are called "open" repeaters and those which have some kind of requirement or permission are called "closed." Closed repeaters are actually pretty rare these days, most repeater owners are more than happy to have anyone use their repeater as long as they are not intentionally being a nuisance.
additionally, VHF repeaters generally are horizon + 10% sort of range, and there is a committee to help with frequency allocations to avoid interference between systems that are close together. So really, the "slice of the band" is regional, the same way (broadcast) FM stations are.
Is this enforced by the HAMs? I didn't think it'd be possible to claim ownership of a slice of the amateur spectrum. Is it just a public shaming?