Federal law was amended on this matter to permit the inclusion of apostrophes in recognition of the large number of Irish immigrants with names including that specific character — until then merely dashes were allowed.
The government does not have a duty imposed on it by the U.S. Constitution. The other foundation that would bind the U.S., and all state governments, would be self-executing treaties. The International Covenant on Civil and Political Rights provides a unified legal framework for many civil rights which, in Article 24(2), provides that
“[e]very child shall be registered immediately after birth and shall have a name”
which, for the purposes of this treaty, arguably should mean that that name shall be recognized by all signatories. This could provide as fertile ground for case law to arise out of for the recognition of, for e.g., diacritic marks or other characters common in non-U.S. names. Even though the U.S. is a signatory to this treaty, and the Senate ratified it in 1992, 16 years after it came into force, it did so by declaring reservations including that "the provisions of Article 1 through 27 of the Covenant are not self-executing" specifically designed to prevent its enforcement in the United States, and to prevent anyone to sue thereunder.
In these regards, the Wikipedia article on the matter cites "Globalism and the Constitution: Treaties, Non-Self-Execution, and the Original Understanding" by John C. Yoo and states:
“As a reservation that is "incompatible with the object and purpose" of a treaty is void as a matter of the Vienna Convention on the Law of Treaties and international law, there is some issue as to whether the non-self-execution declaration is even legal under domestic law”
However, this argument would be found moot by the U.S. Supreme Court as the U.S. also has failed to ratify and enact legislation for the enforcement of the Vienna Convention of the Law of Treaties, or as it is often referred to "the treaty of the treaties".
For this reason, it is highly doubtful that a suit for states or the federal government failing to recognize or use other than 26 letters, dashes and apostrophes any other letters would be successful absent specific legislation setting forth such a duty on a state or the U.S..
Might be worth noting that California tried to pass a bill, but eventually shied away seeing the bill of ~$10M it would have had to foot to change its computer systems to permit for the use of marks and characters beyond the 26 letters and 2 characters.
Conclusively, even if one did, it would be, by all indication, a moot case to make based on the authorities California courts recognize to be bound by.
I also could not find a single case that reached any appellate courts with binding authority in California.
The only state... texas
I found one state to have enacted legislation permitting Latin diacritic marks being, maybe somewhat surprisingly, Texas:
(a) In this section, "diacritical mark" means a mark used in Latin script to change the sound of a letter to which it is added or used to distinguish the meaning of the word in which the letter appears. The term includes accents, tildes, graves, umlauts, and cedillas.
(b) The department shall ensure that an original or renewal driver's license or personal identification certificate issued under this chapter properly records any diacritical mark used in a person's name.
(Tex. Transp. Code § 521.127)