The Center of Attention
It is possible that many people who have realized a person is isolating their partner believe they are the type of person who wants no one else in their life, but they could be very wrong. Those people often hide within a crowd, and they are not necessarily intent on hiding. They can be urbane and outgoing at almost every social event, and they get noticed by many people. The issue is not that they want to be ignored or isolated from anyone, and they instead really want to be the center of attention.
Part of what an isolator does is manipulating their partner so they are the sole person in their life, and it makes them the center of attention when the two are alone. Some people find it difficult to believe, but the truth is more likely that this person craves being the only important being in their personal universe. They have selected one individual to be their adoring sycophant, and it matters little if their partner realizes they are the choice.
Keeping a partner away from others is not always possible in the physical world, so the isolationist works to remove them in the psychological world. Their goal is to have their partner depend solely on them for their needs, and they are expected to agree with each and every aspect of their world view. This gives them the adoration they seek, and it keeps them happy to be the only person worthy for the partner they have selected.
As long as their partner goes along with their fantasy life everything is fine, but when any other influence intrudes the trouble begins. Those who are true isolationists will find ways to abuse or discredit a partner who strays from the path that has been chosen for them in the relationship.