What are friend codes

The friend code system is basically the same as adding someone via username, but instead of that, you use a code. It's largely disliked because of the effort it takes to do something that should be simple. I don't think anyone here wants to remember a 12 or whatever digit code just to be friends with someone online, especially when you both needed to share codes for it to work on the 3DS (only one needs to do it on the Switch). They were removed on the Wii U and came back with the Switch.

why is sharing them on Miiverse agains its TOS

Because they apparently said so? I'm not going to read the ToS to verify if it's true, but if that is what they said, then it is just something they decided for whatever reason.

why are people angry about this?

Some people just want to make friends and the friend code system makes that harder. Many believe this whole thing relates to Nintendo trying to be child friendly, though I've heard some argue it's actually about Nintendo wanting it to be limited to true friends or foster human interaction. Whatever the case might be, it's a tedious system that Nintendo, for whatever reason, seems to stick with.