For example, in french there is the name "Arnaud", which exists in German as "Arno". For a bilingual child it's much more important for the name to sound the same that to be written the same.
There are downsides to the different spellings.
I have this issue within English. There are several ways to Matthew. If misspelled, it is usually Matthew. Occasionally, some spell it Mathieu.
I hate to use phones for any kind of personal info transfer for this reason, as it has caused headaches everywhere from the bank to travel agents to charitable donations to even sharing my email.
For Michael, there is only one spelling, but people nevertheless frequently misspell the name.
My favorite comment on this topic came from a Michael who, when consulted about the spelling Micheal, observed "people named Michael don't spell it that way".
Maybe when they're a child, but once they become an adult, having the same spelling becomes important to avoid bureaucratic headaches, especially now that KYC is becoming so strict. Are you sending money to yourself or someone else when the name is spelled differently on the two accounts?
Arnaud and Arno sound the same! (in this language pair.) In Dutch, I know both an Arnaud and an Arno, and it's pronounced correctly^W like you read it (IPA: /ɑrnʌud/ and /ɑrnoː/) so that threw me off probably.
Anyway, your request is a bunch of human labeling work if there isn't already IPA conversions for every name (and if LLM can't already guess correctly 95% of the time and that's good enough for an initial comparison), but from an algorithmic standpoint shouldn't be hard: use the same comparison but on phonetic spellings of the names rather than language-specific spellings. Example: in Dutch, we pronounce "u" like IPA /y/, whereas German pronounces it as /ü/, so any name with "u" in it will automatically be incompatible pronunciation-wise.
This feels like it will be annoying whenever someone asks for your name in order to write it down or when they are trying to read it. This happens a lot in a school context.
Of course only one of them is your 'actual' name anyway, the other is just by its existence making your name familiar and easy to pronounce. It having a different local spelling (if that's the case) doesn't have to matter or be annoying unless you decide it is. Anything where it's important obviously you make sure to get it right, as you would anyway.
Even simple names, like somebody mentioned Maria, can sound different enough to be annoying in the right parts of the country.