This came up way back. Under Texas state law a convicted felon may possess a firearm in the residence, in which he lives, once five years have elapsed from the date his sentence was discharged. This means the later of release from prison or parole. This is not true under federal law.
Long story short? A convicted Texas felon can legally own a gun in their home so long as they meet the abovementioned legal terms. But if prosecuted under federal law, the same resident can be convicted of the crime of illegally possessing a firearm.
My note: I have no idea whether Alex could legally purchase a gun, (background check) however there is the gun show/gift loophole, and since he can legally possess a gun I don't think the police in AZ bothered to ask.