There is no standard central registry of names in the US. Your "legal name" is whatever you say it is, with the restriction that you may not change your name to commit fraud or try to avoid a debt. The flip side of that is you have very few avenues to force anyone else to encode or display your name in the manner you prefer. If you change your name or have a name that is awkward to deal with, the obligation is generally on you to work out a working relationship with everyone else who needs to know your name.
A "legal name change" is just a helpful service provided by your local government court that lets you publicly declare and record a new name, bound to your previous name, on standard papers that other government agencies and various private organizations MAY (but not MUST) pay attention to.
There are multiple issuing authorities for IDs. Public, private, commercial, corporate, educational, military, government, ... An authority to issue IDs reaches only as far as their legal mandate, and no farther, except by rough consensus and a need to make stuff work.
Each issuing authority gets to have their own regularization rules and database schema. They will pick what works for them, and have very little interest or budget in completely reworking their databases and processes to accommodate someone who is being a pain in their ass.
Whenever you intersect with government, just expect ascii7 smashing.
US banking know-your-customer rules require ascii7 smashed names for account holders, both individuals and companies, to interface to government banking regulators and auditors.
The US interstate Driver License Compact imposes ascii7 smashing and length limits, for database and lookup compatibility reasons. When a cop looks at your DL (which you do have to give him if he's detained you while you are driving a car), it had better match what pops up on his terminal when he enters the the license# and/or the plate#, or you may be are about to have a bad day.
Things don't get a lot better when dealing with passports, from any country.
Passports impose ascii7 (with various accent composition hacks) and length limits, by treaty.
Different countries impose additional different rules and length limits, for reasons various and mostly stupid. Some countries try to appear to still permit accent marks, by trying to encode accents using various slightly different composition encodings layered on top of ascii7, and hoping that everyone uses the same encoding. This kind of sort of mostly works, except when it doesn't.
Passports from countries with widespread non-latin charsets will require an ascii7 smashed name, and often require a specific approved romanization algorithm, or an approved thesaurus, or both. Such a passport MAY have an additional field for the name in a local charset, but that is entirely at the option of the issuing authority.
What is printed on the passport photo page should match whats printed in the "machine readable zone", which should match whats on the rfid chip. "Should".
Most countries require air, sea, and rail carriers to pre-transmit passenger manifest before arrival. The names on those manifests also get ascii7-smashed, and any accent composition stripped, and had better match what's printed in your passport, or you are going to have a rough time boarding at departure, or clearing passport control at arrival.
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Internal and external passports in China used to also require a field encoding the name in the form of Chinese telegraphy numbers, to sidestep transliteration issues between the various Chinese languages. I don't know if it still does.