This is not (universally) true. I've had many friends/family deposit checks for me at my local branch when I'm out of town/on vacation/whatever. I've done it myself for other people. I believe the banks in question were Wells Fargo and US Bank.
YMMV based on bank, but there is no regulation preventing this.
The reasoning presented was a little suspicious - they envisioned someone "forcing" you to unwillingly take a loan (of like $100) by depositing it into your account. Then they would demand their $100 back and use this as leverage to blackmail or otherwise disrupt your life.
But I don't totally disagree. To me, depositing money into another person's bank feels...improper - like performing an oil change on someone else's car, or taking someone else's books back to the library, or walking into your neighbor's unlocked house to lock all the doors. There are very good reasons someone would want these done for them, and also edge cases where these would cause an issue for the other person. It's a moral grey area to me.
Definitely not the case in many (most?) banks. I've done this multiple times and never been turned down.
Banks in the USA do not allow deposits into accounts without account holder permission. They won't let you stop by a branch and deposit into a friend's account (to pay them back) without a formal arrangement, so it's easier just to find your unsophisticated friend and give them cash directly. I'm not sure if this is due to a law, or our culture of security theater.