tl;dr. No. Implementing a system of absolute identification across the entire Internet 1) would destroy the inherently distributed nature of the network and 2) is probably completely impossible anyways.
In it's most basic sense, the Internet allows any computer (and by extension any person) to communicate with any other computer/person on the network. Every client is (or at least should be) treated equally. It's what makes the Internet powerful and what makes it dangerous. It's led to drastic changes in all of our lives.
I can connect with more than a billion people on Facebook… I can find something about anything on Google… I can listen to any song ever on Spotify…
I have super powers.
Peter Parker's Uncle Ben told us all that "with great power comes great responsibility." The Internet didn't create bullying. It hasn't made us worse people. No, the Internet has amplified our flaws. It makes it really, really hard to ignore the things that make us uncomfortable.
We have two choices:
We could remove the characteristics of the Internet that allow activity that makes us uncomfortable, consequently hindering or possibly destroying the power and integrity of the network. We can wrinkle our noses, avert our eyes, and push the problem under the rug.
Or we can embrace our power, this extreme visibility, as a chance to solve actual problems with solutions instead of bandaids. Our new found connectedness presents us all with an opportunity to gain insights into situations that we may never have encountered in our own experiences.
I think the choice is obvious.