I usually hear this claim saying that 'when Christianity expanded, they also demonized the local/pagan gods to impose its superiority'. I used to believe this claim also, but then I noticed some 'inconsistency' in it. I find out that most of the 'demonized gods' are from the Near East, like Baal and Astaroth, but we seldom (almost never) see Greek or Norse gods being 'demonized', like Athena or Odin. Why is it the case, if the claim I mentioned is true? Or the claim itself is just totally unfounded? Hope some informed minds can shed light on my doubt.
This isn't my area of specific specialty, but I think your question can be answered in a superficial way with attention to the Edict of Thessalonica (in the Codex Theodosianus xvi.1.2.) in AD 380), which was issued jointly by the three Roman emperors of the time - Theodosius I (emperor in the west), Gratian (emperor in the east), and Valens (technically, the junior emperor to Gratian). The Edict essentially stated that Nicene Christianity (dating to the final declaration of the Council of Nicea, AD 325) - that is, the form of the religion which specifically believed that the father, son, and Holy Ghost are of the same divinity and same substance (known via the Greek word homoousios). The Edict required many things, but among them that every Christian believe in this verision of the faith or be considered a heretic (literally, 'demented and insane', dementes vesanosque) - and that was among others professing the Christian faith.
In terms of enforcement, the Edict was strict - churches were turned over into the care of bishops who followed the Nicene creed, the right of assembly was stripped of all others, etc. But as for those who were not Christians, the Edict utterly altered their daily lives: temples were ordered closed, no sacrifices - the primary way Greco-Roman religion functioned as a reciprocal connection between human and deity required gifts of thanks or request via sacrifice - but more to your question, that anyone shown to be offering sacrifice "let him be stricken with the avenging sword," (Codex. Theod. xvi.x.4), and that his property should then become property of the city. Even the local rulers were to be punished if this activity was found in their jurisdiction and they failed to punish it. So: the shoe was now on the other foot, so to speak. Surely there are more ways to approach this question, specifically from regional perspectives with more information about the specific ways this was/was not enforced, and how this interacted with local religious traditions, but I think this comes to the heart of your question.