He could just have just wanted to make a statement that he was a protective grandchild because he feels a bit guilty or sad about leaving his wider family to move to LA. If he loves his grandmother (and it sounds like they’re really close) and he doesn’t want to live near the rest of them, he might think they’re not good people to live with for his grandmother too. It could be as simple as that.
He has never lived with his grandparents so it’s probably not about physically protecting her security wise. Maybe he feels her physical care and protection isn’t great because she’s had falls at home or something.
The Queen has presumably been grieving her husband, (and him his grandfather) without having Harry there physically until this point so it’s not about hugs and emotional support. Likely they talk a lot on the phone though. Loss is likely particularly magnified for him, having lost his mother.
Harry maybe wanted to ‘modernise’ the royal family when he thought he was staying ‘in’ (or maybe feels it would protect her if they did that now he is ‘out’)? So maybe he believes her taking part in a Netflix documentary that he controls editorially would do that - and she’s been advised against doing it (obviously!)
Maybe he feels she needs protecting from the British media that he distrusts so much.
Maybe now he’s fallen out with the government over his family’s police protection, he wants to take over Royal security himself?
Protect her from accusations of racism?
Given how non specific (yet very incongruous in a short superficial interview) that ‘protect’ comment was, it does feel a bit like projecting his own issues on to her maybe. But we’ll never know. Could just be a deliberate teaser for his new book!