using Lr on 2 different computers

If you absolutely need this workflow, you may consider Lightroom Desktop (Cloud) because it works with originals in the cloud.

And (for Lightroom-Classic) you might find some useful information here-

HOW TO USE MULTIPLE COMPUTER CATALOGS- (Lightroom Queen)

Why am I not able to just ensure I have internet connection for Dropbox access and move my external hard drive from one computer to the other and have it seemlessly look and function the same?

This is hard to understand. However, if I am understanding you properly, then the answer is: because catalog file or photos in Dropbox causes lots of timing problems, which causes catalog or photo corruption. Not recommended.

I use Lr and Ps on both a windows and iMac computers. I use Dropbox as my "source" of photo location and an external hard drive as the "install to" location.

You cannot install software on an external drive, move the external drive to a different computer, and then expect the software on your external drive to function properly on a different computer. The software would have to be installed on that different computer. This is not something Adobe has done; this a restriction in your operating system which causes this to happen.