This is an interesting question, and I'm not sure I can actually answer it in any kind of definitive way.

The Ladies of Llangollen, for those who don't know, were two upper-class Anglo-Irish women who chose to leave their families in order to cohabit in Wales: Lady Eleanor Butler was the sister of an earl and Sarah Ponsonby was the cousin of another. In March 1778, they attempted to run away together dressed in men's clothing and carrying pistols, but were caught the next day and forcibly separated - very similar to a heterosexual elopement of the time. Everyone around them agreed that they should not have contact with each other and the Butlers intended to send Eleanor to a French convent, but during a moment they were given to say goodbye, they planned a new escape again in April, in which Mary Caryll, a maidservant to Sarah Ponsonby's family, helped hide her in Sarah's room. In the following days, the families agreed to let the women go their own way, and they sailed for Wales. Eventually they moved to the Vale of Llangollen, where they lived in relative obscurity for the rest of their lives (until 1829 and 1831) - locals didn't know who they were, but their story got around and tourists would come to visit.

Mary Caryll's place in the narrative was always uncertain, because as a domestic servant she was generally considered beneath the notice of the intellectual nineteenth-century commentators who discussed the Ladies - she gets referred to as the servant who lived with them, but no attention was given to how she got there (did she accompany them, or was she sent for?) or what her relationship could have been to them. However, we do know that she was considered something more than an employee because she was not paid - Fiona Brideoak (in The Ladies of Llangollen: Desire, Indeterminacy, and the Legacies of Criticism) compares her to a "kin-related female housekeeper", it being a fairly common situation for an unmarried or widowed sister or cousin to move in with an unmarried male houseowner and perform general household management - and was given money for her hair powder (normally reserved for footmen) and for entertaining "the higher orders of the local help". She also was referred to as one of their "truest friends" in a diary entry by Eleanor, left Sarah some land in her will, and shared their tombstone.

It's not impossible that their relationship was actually a polyamorous triad, but there are a number of other possibilities. They may have begun to regard her as a friend when she helped them in their second escape - perhaps they were so grateful for her help that it elevated her in their opinions into a kind of "poor cousin". Maybe she was a woman who loved women, and that gave them a higher degree of sympathy for each other. We can't know - just as we can almost never really know about the interior romantic and sexual lives of historical figures.