Assuming that you've already assigned an alias name to each partner ID, I found that you must be logged in for them to appear instead of just the number. To log in, click on the "Computers & Contacts" button on the left side of the TeamViewer console. Once logged in, go back to the "Remote Control" screen and the Alias should appear.
If you need to assign an Alias, go back to the Computers & Contacts and you can assign them. You must first select the computer that needs an Alias, then select the options button in the top right of the TeamViewer console. A popup with the Alias field should appear.
I tried this. When I logged in to Teamviewer--a requirement that makes me feel extremely uncomfortable, due to privacy concerns--I did get a field under Partner ID that allowed me to an assign an alias. However, when I then listed the IDs (clicking the down arrowhead), the aliases I assigned did not appear, and there was no longer an icon to click to add an alias. I checked, and I am still logged in. So this feature seems buggy; in any event, it is not user friendly. I'm going to just keep my own text file to maps names to the ID numbers I have in my list, rather than log in each time and try to get this feature to work. (Besides, even though you seem like decent folks for making this extremely useful application available, I’m still darn suspicious of what access my logging in provides you all.)