I know the Constitutional rights of service members are not so clear as the general public. But does the Army have blanket authority for such an action?
yes, the can dictate what info is on the ISSUED tags.
Name, social, faith denomination, blood type, etc.
I had my tags wrapped with OD green 550 tape to stop the “tinkle” noise of metal on metal. . .and the chain that goes around your neck, I removed the threads inside a para-cord and threaded the chain through it to stop the metal chain from snagging on my chest hairs.
Good luck trying to unwrap the tags.
The regs I know of (Air Force) did not prohibit wrapping tags, learned about that tip from a Combat Controller.
The backside of my tags: “Blessed be the LORD my strength, which teacheth my hands to war, and my fingers to fight.” (Psalm 144.1)