It is a 'bridleway' because it is intended for use by horse riders. A path doesn't need to be a 'public bridleway' (legal right of way) to be tagged as a bridleway.
See: https://wiki.openstreetmap.org/wiki/Tag:highway%3Dbridleway

As for access tagging, [foot/horse]=designated is correct over [foot/horse]=yes, because 'designated' means there is signage indicating it is for use by those modes, whereas 'yes' means there is a legal right of way.
See: https://wiki.openstreetmap.org/wiki/Key:access