There are also names which have no written form.
Also as far as I remember there are cases where there are multiple "forms" of the same character which theoretically should be purely cosmetic but over time the same name with some of it's characters in different forms has developed different "meanings" which for countries in which the "meaning" of a name matters is a relevant difference.
But as they normally don't mean anything different they have the same code point in unicode and (had when I read about it) no modifier to denote the differences either.
So while this names can somewhat be written they would have to be written in a form which can be seen as degraded by the person having the name and as such forcing them to do so might be seen as mean, impolite maybe even slightly insulting.
Sadly I don't have a reference to the article anymore. Also I don't remember the languages this appeared in but I think it was related to mandarin but I'm not completely sure.
Still the technical overhead of supporting non-unicode is massive and in general not worth it for nearly any application. Exceptions include historical use-cases cases and government use-cases. Also unicode strives to cover "all there is" so it might be "fixed" by now in some form or another.
For most companies I guess you would just ignore it as any alternative would be not worth it by a far shot.
For some special cases using the private use area(s) + a db storing images of the glyph + auto generating a font including glyph generated from the images could be one solution. Another would be to force people to change their names.
For other use cases (e.g. historic) storing a Unicode "approximation" of the name and a picture for the name (full or parts of it) can be a viable alternative, allowing you to generally just have Unicode but still store the information which was discarded when approximating the name in Unicode.
Also trying to extend Unicode can always be a option.
Except that in some cultures names with illegal characters (depending on what you see as illegal) are on your passport. And some other cultures might not have passports.
The simplest reason is that not all names can be mapped to unicode.
Which for a lot of software don't matter as supporting non-unicode names is not worth it.
But if you are a government or some system used by one, or some other institution where you can't just randomly turn down some people from being registered in your computer system this is a potential problem.
Lastly there are systems which can contain fictive or historic names so じゃわØÆÅ?+ might very well be a name.
The only sane way to handle names is to handle them as opaque blobs which you display and maybe do a (fuzzy) search on but not much more.
Never _ever_ use names as any form of identificators in _any_ computer systems as they are both highly imprecise, unstable in time, unstable in context and unstable in interpretation.
(E.g. the same glyph sequences might be the same name or not depending on some historic context specific to the person using the name including weather or not the name is used in a way where it is "imported" into a culture, how people might have moved between cultural context in their live time and the live time of their ancestors and other factors. This is so extremely specific to the person using the name that there is absolutely no way to handle this (besides asking the person in question)).