> Although I don't support it, would the internet be a better place with people being identified?

I don't believe so.

> For this to be a constructive discussion, can we assume companies have a proper and secure way to validate identities and store your data?

What?! No, we absolutely can't assume that. In fact, good default is to assume that data stored will be eventually compromised with high probability. Assuming otherwise is like believing in some magical backdoors in cryptography that only "good" guys can access and other such nonsense.

> I sometimes wonder if the internet as we know it and social medias pervasive influence would be much different if people were less anonymous.

Social media as we know it today mostly stems from FB which has a real name policy. Don't think FB is much better than others (by any measure that interests me at least).