Indeed it sounds like the ole good battle raged between the prescriptive and descriptive grammar rules.
Many (I think) would say that the way to say that is "The lost dog responds to his name, Xena" or "Lost dog: responds to the name Xena." etc.
In a prescriptive way, one more prone to usage as it is, would say okay to that sentence. Or would just not say a thing, because it´s a established way.