Is our relationship unhealthy?

It's a choice to evaluate partners based on how your family likes them. In this case, your sister doesn't like your partner because of past events that you don't mention, so she probably has a reason. So for now you should tell the person who told you that it's unhealthy that you'll consider it and change topic.
For your sister to cut contact with you it's a huge step, and we would have to look at the context to see if it's reasonable or not, but in general, if your partner isn't ruining your life and harming you, she should stick by you and just politely avoid your partner, so relax. I suggest you regularly talk to friends to have an external opinion on this, so you can be told if anyone is doing specifically unhealthy things.

I have a similar situation. I have a girl I consider my sister we have been friends over 16 years. There was a time when I had a bad breakup and my sister was there for me. She ended up destroying almost every picture or thing this ex of mine had given me. She wrote her a nasty email telling her off! Later on my sister would have a boyfriend that I didn't really care for and he was jealous of me. Although my sis and I were just friends he didn't like us hanging around. Well eventually they would go on to move together and they had been together quite a long time. I realized this may be the guy she spends the rest of her life with and she seemed happy so I took a step back and let happen whatever happened. I NEVER tried to get between them or make her feel like she had to decide between him and I. I tried to be the better man and just take a step back. For a few years we didn't see each other at all and only kept in touch online for the most part.Well eventually the relationship ended because of the way he really treated her... like a possession and now we are back together. She vowed to never let any guy ever get between us like that again... now we've been friends over 16 years it is not likely any guy could... never the less. If your sister really loves you then all she really wants is for you to be happy... right now it seems she is fighting that feeling and the feeling of thinking she knows what is best for you. Parents often do the same thing. If your sister really loves you then that love should be unconditional and no matter what... so long as you don't close the door on her she will forgive you and accept whatever your choices are with this girlfriend of yours.

I think that it depends on the reason she wants you to break up with this person. Is it for your own good or for her own good? That's the biggest question, and if it is because she just wants to be controlling, then it might not be a great situation. You said that you were with this person and it ended badly, and I know that if I got back with my ex or my sister got back with hers, we would be upset with each other about it because we are protective of each other, so in that aspect, it isn't unhealthy. It would be good to talk about it and think what is best for you.

LEAVE YOUR GIRLFRIEND, AND LIVE THE LIFE YOUR SISTER WANTS IT. YOU SEEM TO BE A GUY WHO'SE A PEOPLE PLEASER. I PITY YOUR GIRLFRIEND SHE DOESNT DESERVE TO BE WITH YOU.