For example, I came across this YouTube video in which the author tests a whole bunch of batteries and displays a series of bar charts showing each battery’s performance in various categories. How do content creators capture and log battery output like that? Is there a specific tool that plugs into a PC or records to a MicroSD card or something?
Yes you need a specific battery tester.
Here is a Video explaining how to test batteries: How to Properly Test New and Used Li-Ion Batteries (Including Cool Testers)(15:12) by Andreas Spiess
At 6:14 he recommends a few testers. The more expensive ones list connection options via cable and Bluetooth.