Cinders, I've been a step parent most of my adult life and it is absolutely NOT normal to call the 'steps' names reserved for original family.
My step-kids call me and my parents by their first names. I've got heaps of friends and family in blended families and I can't think of one who calls the 'steps' anything other than their real names - except where there are very specific family names - like everyone in the world calls my x's mother Mama - so my sisters and their kids call her Mama.
You H is trying way too hard. He's trying to do what he thinks OW and her family want him to do and it's unacceptable. Frankly, I suspect they would be a bit uncomfortable with it too. They would have to feel a bit put upon. It's weird.
It's OK for you to tell him you think it's weird or unacceptable that he wants them to call the OWs family by such intimate names. Perhaps they could come up with a polite nick-name - but only if that is completely different from what they call their real grandparents.
You are right to be angry. The man is giving away stuff that's not his to give away.
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Never make someone a priority, who makes you an option.
