No, I don't think its legal to have a pet cardinal.
If it is very important to to you have the legal authority on your side I'd contact the local wildlife agency or else ask a local vet for assistance. If your city has a local information line (in New York we have 311) you can call them and ask for the proper agency to contact.
The following answer is from a person who is a
Lic. Wildlife Rehabilitator, 17 yrs
Avian & Small Mammal
Forner Dir DRS Animal ER
You bet it's illegal. How are YOU teaching her to fly?
Her parents have to be frantic. It's sad because fledglings are the victims of too many people who pick them up and decide to keep them. The bird is miserable and lonely without the company of other birds of their own species.
ALL fledglings are tame and submissive at this stage, but she will not remain so. She will always want to be free no matter how much attention you give her.
The bird's mother may still be around. Put the baby outside and step away. Give the mother a chance to find her baby. If you are positive after a day or so that the mother is not coming back, then you need to find a licensed wildlife rehabilitator. There are plenty in Texas. It's the right thing to do for this little one. Her requirements are much more than a dish of seed and a bowl of water.
If you love wildlife, volunteer for a sanctuary that cares for them. Please don't cage that cardinal. It has every right to live it's life in freedom.
Let Nature take care of her own.