Discovering a relative

I live in Pembrokeshire, West Wales, but was born and lived for my first 30 years in Swansea. I have a daughter who is a teacher in a boarding school in Shrewsbury. She is also an Asst. Housemistress at one of the boarding houses. This term they have a new Matron. Matron has a dog, as does my daughter. I was visiting my daughter last week, and took her dog for a walk. He's a very friendly dog, and always wants to say hello to other dogs. We saw a woman with a dog, and the two dogs greeted each other and started boxing. I called my dog's name, and the other woman realised it was my daughter's dog, and introduced herself as the new Matron of the boarding house. We decided to let the dogs play, and had a chat about her new role, where we lived etc (she was from Liverpool, but had lived all over the world). She had to go back, and I was going to continue my walk, but then decided I'd walk back with her. She then asked if I'd always lived in Pembrokeshire. I told her that I was from Swansea, and she replied that her mother was from Swansea. I asked which part, and she said her mother was from Bonymaen, which is where both my parents lived. I asked what her mother's maiden name was, and it was the same as my mother's. I was taken aback, and said we must be related somehow in the distant past. She started saying names, none of which meant much to me. I told her my mother's first name and her siblings. I then told her my grandparents names, Morgan and Janey. She stopped and said that her mother had an Uncle Moggie ( which is how he was known ). My grandfather committed suicide so I told her that, and she said, 'yes, because he thought he had cancer', which was the reason he took his own life. It took a few hours to work out, but her grandmother andy grandfather were brother and sister, and we are second cousins - totally unaware of each others existence. (Her grandmother married someone who also had the surname Vaughan, hence why her mother'saiden name was. Vaughan)