I have been thinking about installing Linux (probably Mint) on a bootable external eSATA SSD. Unlike a live CD, this would not be a canned setup that would remain unchanged and would therefore function in the same generic manner on any computer to which I might connect it. I would plan to configure and refine this installation as I went along.

My question: what happens if I boot this SSD on one computer, adjust it as desired, and then shut down and boot this SSD on a different computer? Example: if I install Windows on one computer, make an image of that machine's system drive, restore that image to another machine, and then boot that other machine, I can have all kinds of problems. Same thing in Linux?

You might have problems, depending on the level of customization :
Most important, the hardware. The configurations you make might break something. Don't know how much you could keep that generic. You could have a look at Alien Bob's live slackware image, wich is well documented.

What do you really have in mind ? Switching between a few known computers or having a custom distribution usable on every computer you might use ? If it is the second option, there are live distros out there that support persistency (live slackware do).

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For most modern linux distro's on most modern computers there is no difference between an external drive and an internal drive. You install a distro or more to this external just like an internal. Most settings will remain constant over compatible hardware. A few issues may occur with how video works and maybe you will have to play with network settings. Each distro acts somewhat different on moves.

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If I understand that statement correctly, you would be violating the EULA from microsoft as you are paying for privilege of installing that software on one and only one machine. You are allowed to make a backup as I understand. If the original hardware dies or is problematic, you an put it on another computer but I don't what hoops you would have to jump through to do that.

You can't install windows to external drive with their installation medium for obvious reasons. The primary problem you would have with a Linux system is if you install proprietary drivers of some kind, particularly for graphics cards and then move the external drive to another drive. That's one example.

Depending on the windows version it may cause quite an issue. I put windows 10 on a few flash drives and they do OK. Some hacks to windows 7 back to XP let you make it portable.

When linux first came out you basically had to configure it somewhat like linux. It was not easy to move it to a new system as the installers of the day and other features of linux made it rather locked to a system.

Slowly linux became able to be moved quite easily. Installers left things generic and great support was built into "distros".

Sadly some of the older moving system to system problems are coming back. Issues with video mostly but some drive enumeration and some networking issues.

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