Usually, tame is used to refer to animals which have been domesticated in some way. For example, a tame dog won't attack its owner and might even help with hunting and so on. In this case, tame is used to suggest that the emperor Augustus has 'tamed' Horace, such that he doesn't attack him in his poems, but instead writes what he (the emperor) wants. In a way it is slightly demeaning to Horace to be compared to an animal with a master.
"Obliging" misses the notion of "coming round later" as Thomas Tompion puts it, that can be implied by tame. Something that is tame is potentially wild, or intractable, but has been made "tractable", usually by good treatment or favors.
Horace was once hostile to Augustus.
Poets as a group are often thought to be people who are, or should be, free of the constraints that bind the rest of us.