What does 5G E mean on my iPhone 13 with current update #16?

It means you likely have AT&T service or service with a company that uses AT&T's network and are connected to their "5G Evolution" network - meaning it is actually 4G LTE with some 5G enhancements, not a true 5G network.

If you do an Internet search for "5GE" it will show a number of articles from third-party websites that will explain it further.

Here's AT&T's explanation:

https://about.att.com/innovationblog/2019/04/5g_evolution.html