Why did Drew have his step dad's last name?

Legal reason is most often the case. Don't really get it, but I've asked real people in this situation, and that's the answer I get.

"Your son is a murderer"--Conan O'Brien--6/11/09 Tonight Show

Gus most likely adopted him. Hence why he keept pressuring Drew about his weight and obeying him. Disgracing Gus' name and making him look bad since he is the sheriff was not good in Gus' eyes.

Work hard. Play Harder!

Because the mom wanted Gus to be a father to Drew, so I'm not surprised Gus adopted Drew to please the mom.

It happens all the time, if a parent remarries and the other parent isn't in the picture it's perfectly normal for the step-parent to legally adopt the child. That way if something happens to the 'real' parent, they'd already have legal custody so there wouldn't have to be a big legal issue, or custody battle if the 'real' parent dies, or is unable to take care of them.
When my Aunt married my Uncle (now ex) he had a son from his previous marriage and she adopted him, and even after they got divorced she still had joint custody of him even though he wasn't her biological son.
They could have also just changed his last name, and Gus may not have adopted him, it's fairy inexpensive to have your last name changed, then they'd have the feeling of being a family.
I know different family dynamics and divorce are pretty normal these days, but 20 years back in a small town, it would be weird to have a different last name than your Mom, especially when your real Dad isn't in the picture.

In this case I think it has a lot to do with Gus's perception of himself as the head of the household. Contrary to what others are saying here, step parents adopting step children rarely happens. Adoption also makes them legal parents and Drew was quick to point out to Jeff that Gus was his step father. Gus was also quick to refer to Drew as "your son" to his wife when he was angry with Gus.

My guess is that as Gus felt he had to be the unquestioned law and head of household that having Drew take his last name was a way he could claim authority over Gus. He obviously had no real feelings for Drew so I don't think it had anything to do with him adopting Drew.