Does AirPort Extreme Show Device Names Connected?

No.. apple actually removed that functionality in the upgrade (so called) to v6 airport utility about 3 years ago.

Let me add the situation is rather complicated.

The clients shown are only wireless.. no ethernet are shown.

They can appear by Name.. by IP or by MAC address and there is no way to change it.

So you will see postings around here which show all three at the same time.. some IP some names and some MAC addresses.. This is due to Apple DNS issues.. of long standing I might note. It can change from day to day. One day it shows name and the next day IP. Roll your mouse pointer over the client name or IP and a box will show up with some more info in it. It is possible the name will be there.. but maybe not.

We tend to recommend using Fing which is a free application on your ipad or iphone.

Although even I do not understand all the info it shows which is different to what is expected.

If you have an older computer or are running any version of windows.. stick v5 utility back into the picture for a real listing of all dhcp clients..

I run both so I can find the listings of everything.

Is there anyway I can see the names of connected devices instead of just a load of numbers?

It's a roll of the dice. Today, I can actually see the names of wireless clients that are connected to the AirPort network.

A day or two ago, I saw only IP addresses. Sometimes, I will see the MAC Address.

Tomorrow will be another roll of the dice.