Yes, this happens, sadly. Maybe their feelings of having been betrayed have got more to do with them than you, but still.... Some significant others do feel like that. Especially spouses.
When I started transitioning, my then girlfriend (now ex) told me she felt betrayed in the way we had been making love. As it was classic heterosexual love, very vanilla. Even though it was something I did, it never really felt good, to the point that I was consciously betraying myself when sleeping with her. Of course after some time, the situation became untenable and I was forced to come out.... In the beginning, she was very respectful and sweet, but as time progressed, she became ever sourer. Asking questions about me lying to her, and at a certain point I also had to tell her I had been betraying myself as well.
In the end, most people have (very) firm beliefs about gender, with it being immutable and binary. Gender serves as a basic category that helps people to make sense of a complicated world, and shaking this 'immutable truth' up will make some people feel betrayed. It starts with your own self-betrayal, and after coming out it spreads to others. What matters is that these are BELIEF SYSTEMS being challenged, it's similar to a child finding out Santa isn't real. Something very solid and tenable suddenly becomes flexible, loses its rigidity and has to be questioned. This then requires lots of energy, empathy towards the people involved and frankly: a confrontation with one's own fears and doubts about life.
Because the binary system still is a stereotype. It's a construct that people cling to when wanting for life to be 'simple', for whatever reason. This construct gets fed by their own fears. Hence the feelings of betrayal.
There is no easy way to go about this, and to make things clear for you: some of us have lost contact with their entire families over these issues. I have lost my girlfriend, with whom I had dreams to have children, settling down and living a long life together. Don't get me wrong, there were other reasons involved and she's anything but a bigot, but still. What matters in the end, is that you stay true to yourself. Don't betray who YOU are, that's far worse than shaking up somebody's rigid or old-fashioned ideas about how life ought to be (in their eyes).
Chin up! 🙂