Yeah, the political or otherwise ideological nature is read into the work. It doesn't come out of the work. But, I think, precisely because PL is one of the greatest masterpieces of the English language, it becomes a focus for that kind of attention, where people want to construe it for their own objectives. See also the centuries-long debate running through the likes of William Blake, C.S. Lewis, and Philip Pullman on whether or not we should see Satan as the sympathetic and virtuous hero. That debate is a touchstone of religious ideological conversation.

As an aside, I find that PL fills the blanks in the biblical narrative almost too well! There are some aspects of our modern cultural understanding of what angels, demons, and Satan are like that come straight out of PL and have no foundation in the biblical narrative. It leads to a lot of confusion among religious persons because they end up believing in these details that don't come from the avowed authoritative source.

You may have heard a Christian try to claim that Adam ate the fruit after Eve did, with the express purpose of not being separated from her---that he knew she'd be exiled, so he intended to stay with her by being exiled himself. In that way, Adam acquires a sympathetic and heroic tint in his role in the Fall. But this is completely fabricated by Milton! It's straight from PL and nowhere to be found in the biblical narrative.