What does it really mean when someone says, "You're their other half"?

Anonymous
My boyfriend has been being really nervous and fidgety lately. He started doing this after I accidentally saw a Tiffany and Company bag in his closet when he opened it. I told him one day I got my nails manicured he got weird by looking at nails being shaky nervous suddenly. Then later on he kept holding my hand just rubbing my ring finger on my left hand. It has been rather interesting and my boyfriend keeps claiming he hasn't gotten me a gift yet.

Then I went over to my SO's sister's for some food since she invited the family over. I went over there and she was super friendly along with welcoming to me as always. I said I am sorry if you just invited family to come over today, but I was already with your brother hanging out. She said, "You are always welcome since you are his other half." Then his sister was telling me I should come over in the summer for Margaritas with the girls and a BBQ sometime. Otherwise asking me if I liked something so we could all do a girls night involving; their Mom, Sister and maybe their cousin.

Why would his Sister call me his other half and why might they both be acting like this? Is it definitely possible they know something I do not?
What does it really mean when someone says, "You're their other half"?
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