I think that is super irritating but not worth causing a big drama over.
If you are creating a stable home Dss will recognize that and feel at home.
I have seen this play out in my own - albeit very different situation. I told xh for years that it was up to him to make his place a home but instead he focusses tons of attention on her with crazy demands and always in competition with us. We have never competed, just tried to make sure we are a family. Where do you think she feels most connected and at home?
And it's not just about custody. My step siblings felt the same way about our home and they were only with us every other weekend.
My brothers kids are the same way and they split their time until the oldest refused to go to her mom anymore.
DSS will be best served by keeping things as civil as possible so my advice is to focus on managing the crazy ex as best you can by choosing your battles, and helping DSS understand without revealing too much of the adult tension (sounds like you already do this).
I would ignore this and just explain to DSS that his mama meant that mom and dad will always be together in their love for him but that they won't be living together, that he has two homes now and two families to love him etc etc.
anthropologist mom to O
(12-30-04) and R
(4-22-10) Married to my best friend