Your child has both names as his surname on his birth certificate, therefore his name is legal
However, I notice that although you are married you don't use the title Mrs either, preferring to be addressed by your maiden name, hopefully not prefixed with the ghastly Ms. Making a point? Not sure what.
As a teacher I found that two surnames always created problems: wouldn't fit into spaces on registers and official forms, name cards, tray labels, books, clothes etc. The children often disliked them because they wanted to have one surname like everyone else, and frequently dropped one when they reached an age where they took control.
Then there was the problem when the marriage/relationship broke up, and a parent tried to drop one of the names and replace it with that of the next partner.
You say you don't lose sleep over it, but you care enough to garner support here.
My sympathy is increasingly with your mother in law.