I'm super excited to see this and I'm hoping there's no odd distance limitation. My friends span the globe and I'd love it, if this is a good way to get new and interesting postcards.
It works be great to see mushrooms create opportunities to create friendships too. One thing at a time, I suppose.
You can friend someone by using their code. It's a sequence of numbers that aren't obviously related to the person.
But you can't add a friend by putting in their name. I think this is Niantic's way of making sure we aren't spammed by invites by people with less-pikmin-oriented goals.
I'm going for a way to friend other local players and that's not easy to do.
Oh! You’re right. I was thinking you could add someone from a mushroom but you can only react. It would be nice if there was a toggle to allow yourself to be friended. I can understand why there might be privacy concerns and you might not want people to be able to add you, but they could make it an opt in system. I see the same people all the time, would love to be able to interact. 
What you describe sounds perfect and relatable. It's weird seeing people on the same mushrooms daily, seeing us obviously cooperate and help each other... And we can't become friends because I have no idea who they are.