Within the EU, there are individual national laws on civil unions/partnerships, but there is no EU treaty or EU wide law on it.
While in certain things there can be leeway between written policy and actual practice, in most others there can not be.
If a country does not accept a civil union, then the couple can not immigrate to another country under such a visa. They will need to find another type of visa or not move, separate or whatever.
Before the UK had civil partnerships, partners were not guaranteed to be able to visit their partner in the hospital. State hospitals {NHS} treated gay couples as not family, while private hospitals were much friendlier to gays. In 2002, I was supposed to have sinus surgery in an NHS hospital and on the admission form they clearly said that all non family visitation was not allowed for short term in-patients. Thankfully, my surgery was pushed to a private hospital due to scheduling conflicts at the NHS hospital, meaning that my partner could visit me the whole time.