The only real way to handle it, even though it's not a fully general solution, is to ask the user if it needs to be different than what you've said is their canonical name. For example, just ask "How should your name appear in a address?" (only if different from your name) or "How may your name appear in government records?" (for when you need to look these up, like for known traveler info). That's not fully general but it at least accommodates many of these use cases and doesn't run afoul of the "automated rudeness" problem the blogger is talking about.

Surely every alphabet/character in the world is a much smaller set than every language though? Say, roughly Unicode sized?

The scale of "all the postal mail in the world" is about as large as it comes, so to a first approximation, every little potential issue you can imagine, as well as every one you can't, will actually happen. Is that "fragile" in Russian or "urgent" in Mongolian? Is that bit of text the oblast or the road?

But even beyond that, even expecting every postal system in the world to have to understand every script is itself not feasible. I don't mean this to offend, but Arabic is just a scribble to me. I know from reading articles about the difficulty of typography that Arabic letters are generally strongly affected by their preceding and/or following characters, but that only from HN, not from my normal day-to-day experience. I couldn't even parse it into letters correctly or safely. The ideographic languages have no spaces, so I can't break them into words safely. (After some non-trivial study of Japanese, I could mostly do it, but still only mostly.) It's not a reasonable expectation to put on every postal system in the world.

I agree that potential integration with non-Unicode systems is a real concern I hadn't thought of. However, your examples of complexity in the service of demonstrating why a system might not support Unicode don't seem to affect or be affected by the allowance of all printable Unicode characters in digital representations of proper nouns, unless I'm misunderstanding.

I.e. I don't understand why these problems mean that a postal system might elect to not support all scripts in storage (I get why it means the OCR software for a postal office might not support scripts outside their own country's scripts).

Because having the script in storage is not enough to derive the other possible representations that may be needed. partdavid's answer is correct, you may have to ask for other representations to be provided to you.

This is much like the naming scenario, where I may need both your legal name for legal reasons, which may have local restrictions on it, and I may also have a field that is basically "What do you want me to call you?" which may have arbitrary unbounded Unicode for all I care, I don't care if my system calls you the Lord and Master of Zalgo Text. But there's no programmatic way to get from the latter to the former, or indeed, even a human way.