​WinCatalog does not require any special installation (installer just copies necessary files to your computer). To use WinCatalog as a portable application, please just copy the entire WinCatalog folder from "Program Files (x86)" to your external hard drive or to a flash drive and run WinCatalog.exe to launch the app.
However, you should know that WinCatalog is not a real portable application and it stores the settings like a window position and the registration info in the Windows Registry. That is why you will be asked for a registration key when you run WinCatalog first time on a new machine (just once, on every next run it will use the stored registration info; and we do not limit you in a number of computers where you can use WinCatalog).
You should also know that WinCatalog uses SQLite database engine in the core and SQLite requires a lot of write operations when you populate your catalog file. Storing a catalog file on a flash drive may reduce scanning speed because usually flash drives work slowly than hard drives (even slower that the external drives).
Summarizing the above: yes, you can use WinCatalog on a flash drive, but the program's speed may be slightly reduced. And it is recommended to keep the catalog files on an external drive connected via USB 3.0 (in order to increase the populating speed).