How can LO get an ID card without a proper birth certificate?

First of all, try again. This sounds like a mistake.
Secondly, what birth certificate has she been using to this point? She has had a license all this time? This isn't making a whole lot of sense to me. Are you saying she has been without ID her entire life and now needs it. She would have had to show her birth certificate in the past for any number of reasons for getting ID. I am very confused by all of this and am afraid I can't advise. I hope others have an answer for you. She may need to see an attorney to help her with all the ID she has had for all her life, and a way to make a "also known as" documentation that is legally done, but I really haven't a clue how this could be done. What a mystery.

The same thing happened to my grandmother back in the 1940s. Her not-very-honest mother never told her that her father was really her stepfather, and she found out when she had to get a copy of her birth certificate to work in an airplane factory during WWII.

I don't know when your mother got her first driver's license, but I'd bet a birth certificate wasn't required when she got it like they are now.

Unfortunately, this is something for an attorney to handle, I'm afraid.

https://www.oflaherty-law.com/learn-about-law/what-happens-if-the-name-i-use-is-different-from-the-name-on-my-birth-certificate

"Your best course of action for the long-term is probably to have your name legally changed and then update your birth certificate. In order to legally change your name, you must file a Request for Name Change along with a few accompanying documents with the clerk of the circuit court in the county in which you reside. This process will require you to publish notice in a local newspaper and attend a court hearing.

Once the judge signs an order granting your name change, you can then submit the name change order along with an Affidavit and Certificate of Correction Request to the Illinois Department of Public Health in order to correct your birth certificate.

At the end of this process, your birth certificate will match the name that you have been using. While this is certainly the best long-term solution, the downside is that it takes some time. The name change hearing will typically be scheduled no sooner than 8 weeks from the date that you file your name change request with the court clerk."

Even if she doesn't live in Illinois, I'd guess the process is very similar to that in other states.

I am assuming grandmom is no longer around. In my instance, I was born with one name. My Mom married again when I was two and her husband adopted me. I didn't know until my early 20s that I was adopted. My Dad had always been my Dad. When I went to get a passport my Birth cerificate was questioned because of the date issued. I then had to send in my adoption papers showing the change of my name.

I am with MJ. Mom was born under one name but went by grandmoms husbands name without going thru the correct channels. Moms name could have been legally changed and not adopted. What birth certificate did she use to get her SS?