Yes, it does. Though:
The waiting time refers to the time the taxi spent waiting (or moving very slowly) in traffic and lights (after the flag is down), but not the time it wait for the passengers to turn up before the flag is down, and
It doesn't cost much as compared to how much the taxi charge when it moves.
The department of transport, Taoyuan City ...
There are a couple businesses within Taiwan Taoyuan International Airport that provide printing / photocopying service.
7-eleven / ibon Kiosk in Terminal 1
The 7-eleven in Terminal 1 (B1) has two ibon Kiosks, a self-service machine that supports printing and scanning of documents (instructions here):
You can print out any documents or pictures saved in ...
According to TripAdvisor, the 7-11 convenience store has printers where you can print, scan and fax.
I can't really confirm based on their website (in English, there is a Taiwan website too) because it's not explicit on there but there is a phone number to the terminal where the store is located that you probably can call to be sure.
That looks like it should be the cost for being stuck in traffic. It's not specific to the airport, just a common way taxi fares work to balance distance and time.
From this government site (mirror) (numbers appear to be out of date, but it illustrates the general idea):
(2) Prolonged metering: It’s NT$5 for every 100 seconds for slow-moving taxis at ...