It's easy to add people with the old YouTube profiles as friend, as you just click the link:
I see no such option where the other links are on the new profiles though:
Is there still a way to do that, or did they just remove it?
They seem to have removed the button, but there is a workaround.
Copy the username of the person you want to add as a friend (just under Created by; 'GamingSE' in your example), and then go to your Inbox > Address Book. Click on New > Contact... and type in the username inside the Username box on the right pane. Now click Save, select that user, click Add to Friends on the above bar and you're done. This is the only way to add someone as a friend at the moment.