Actually, you don't need a Real ID yet. The requirement has been delayed because of Covid for another 2 years. A normal driver's license is still valid to fly domestically and enter federal buildings/military bases, etc. And if you have your passport with you, you won't need a Real ID to fly, even when the rule goes into effect in 2023.
There is no reason Harry has to have a Real ID to fly, or exist, in the State of California. His passport is sufficient to fly domestically or internationally. He doesn't need a California Driver's License (Real ID or not) to drive in California if his U.K. license is currently valid.
And if Meghan or anyone else in California wanted to change their name legally, even before Real ID they had to go to the DMV to do so with documentation. So she likely has kept her maiden name legally, (assuming this TMZ document is correct), or because of Covid, like many of us, has been putting off going to the DMV to make the change. I know someone who didn't bother changing her legal name until 3 years of marriage, because of how much she hated going to the California DMV.
Personally, I think this entire thing isn't a big deal. If the birth certificate application is rejected, Harry and Megan will be required to go in person and fix it, and then it will be fixed and issued.