Hi XXXXXXXXXXXX~
My own dog does the exact same thing...follows me from room to room. I'm not so sure it's anxiety, but rather a genetic need for a dog to be with the rest of his pack...and you and your family are his pack.
As long as he's not destructive when you leave him home alone, and doesn't get whiney or nervous when you're not there, I don't think I'd worry about him tagging along with you in the house. There aren't many dogs who don't do that...so as long as it doesn't bother you, and he's perfect in every other way...I say let him follow.
I hope this helps.
Certified Veterinary Technician
Associates Degree in Veterinary Technology. State and Nationally certified.
That does sound like a typical boxer!
He sounds like a good dog...and as he gets older, he'll start calming down faster..he's still a youngster, so you've got a little time before he's completely calm!