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have not seen the show but isnt that his wife
Why is she embarrased
She is his "wife." In effectively East Germany at the height of the cold war, where anyone could be suspected of being a spy (Loid is a spy), they rush a marriage for convenience. Loid gets a family, which makes him look less suspicious, and Yor is an assassin, so her reason is also to avoid suspicion. They're not genuinely in a relationship, although in the anime they are kind of settling into the idea.
Edit: oh wait I'm like 40 minutes late to this there's already another comment, oh well have another comment
Sort of. They're pretending to be married and each one thinks they're using the other. They both have secret professions and so it's in their interest to look "normal" by having a family so they agree to pretend that they're married to each other. No one but Anya is aware of any of the others secrets.