The Internet is a network of networks. Individual networks may have a database of their own users and their addresses. Some do not.
There is no standard by which one can get (or be) registered. There is also the ``privacy issues" to consider (many would argue the telephone directories are no different than email addresses. The main differences is that the phone companies have had 80 years to get together and collaborate. There are also a lot fewer telephone companies than there are Internet Service Providers + Organisations).
There are a number of ways, within UNIX/Linux, one can use to try to find out information about users. The most often used are the finger and rusers commands. But because of security issues, many systems disable the use of those services.
On the systems maintained by the Research Computing Support Group, the phone command is available. That command is mapped to
whois -h gold.sao.nrc.ca person_name
One may also use Web Browsers to find information about a specific person. Use a search engine, and enter the person's name in quotes. Results may amaze you!