Adobe's license allows for two concurrent activations of Lightroom Classic CC, so you can do what you wish. Keeping your photo library and your catalog on an external disk will allow you to work on the same catalog and images on two separate computers, although not at the same time.
On the second machine, launch the Adobe Creative Cloud manager and under Apps, find Lightroom Classic CC and choose Install. That will place Lightroom Classic CC on the second computer. When you launch it, you will need to login with your Adobe ID to make it active.
Can I have Adobe Photoshop Lightroom Classic CC on two devices and use a single external drive that i switch between computers?
Yes, the Lightroom Classic CC license allows this under certain limited conditions.
Now, if I can figure how to make Adobe let me instal Adobe Photoshop Lightroom Classic CC on my second device!! I do the download and after instal I only get Lightroom CC!
Obviously, if you get Lightroom CC and not Lightroom Classic CC, you have downloaded the wrong installer file. Just download the free trial of thePhotography plan for the 2nd computer and install that, then sign in with your e-mail.