I was doing some test and mostly the plugins works perfect. But when I gave tag to random default plater without op or permission admin he was able to change the displayed name of the horse with tag. When using /zh info it would still show the old name.
// Btw I wish I could disable the 'default name' and random name on claim.

@theadisim I'll investigate the issue with name tag. It should be handled by the fact that ZHorse filters right clicks of unauthorised players but this might not work anymore with latest Spigot/Paper builds.
Please update your issue to include all the information required in the issue template.

// Btw I wish I could disable the 'default name' and random name on claim.

This is normally done by default. Check the config options HorseNames.give-random-names and HorseNames.random-names.