Have been looking at this myself as off to lake Garda in a couple of weeks. Blue badge seems to be region specific with nothing useful online that I’ve been able to find to say which regions may accept a UK blue badge or which don’t. The free or discounted entry for a disabled person or an essential companion seems to be based on a very dated by UK standards assessment of percentage of disability. Free entry to public places or transport if you are 70% to 100% disabled. Local proof of eligibility required. In practice it might mean that you get the same without evidence. Or it might not.