FWIW, I sometimes order items from (most likely) the same online retailer on behalf of other people because I get free 2-day shipping. They tell me what to order for them. Not saying that was necessarily (or even likely) the case here, but whatever. I also used to buy gifts for my last ex's kids on his behalf because he was broke.
Your son is (unfortunately) probably pretty used to dad's dropping the ball on these rare occasions, and he probably doesn't care that much about whose name is on the account--he might have just been momentarily thrown that dad's name didn't come up on the account.
Yes, signing his full name under "Dad" is weird, but a lot less weird than the rest of the neglect of his son.
Glad your kiddo has a mom looking out for him.
Hugs!