Most people wouldn't really have a concept of what a "state" was.

The concept of state sovereignty wouldn't come about until (approximately) the Treaty of Westphalia, but Romans of this time period would have certainly understood what the government-prescribed religion is

At this point the Roman Empire was Christian. The Emperor was no longer considered a god. They were now considered chosen by God to lead the faithful on Earth.

I would not assume that our modern understanding of Christian orthodoxy totally applies. Christianity is a syncretic religion nearly everywhere.

Since it is in the context of a panegyric, it should not be taken as a literal theological statement, but as poetic language. However, I would agree that it is some reverberation of the emperor cult, even 68 years after the Edict of Thessalonica.

Note also that Roman emperors could be given the epithet "Divus" ("the divine") after their death well into the Christian era. The German Wikipedia has a list: https://de.wikipedia.org/wiki/Divus Although Theodisus II is not included, there appear two East-Roman emperors after him: Leo (†474) and Anastasius (†518).

If it was purely poetic language, why the Roman objection to Atilla being compared to a God? Feel like there has to be more to it than that.

The "dispute" was about status, not a simple ontological fact. The Roman did not grant Attila the same rank as Theodosius. Calling him "God" means that Theodosius has the highest rank possible, but Attila not.