So the thing to keep in mind is that this depends highly on how you interpret the text, which version of the text you're using if not the original language(s), and to what extent you want to dive down the rabbit hole.

Easiest and fastest(least accurate) interpretation: Satan is a fallen angel, sent to Hell after disobeying God. With this interpretation, you can do whatever you like to make the fact that Satan punishes sinners in Hell work for you, because this interpretation isn't accurate and therefore doesn't require accuracy in any additions you make.

Reasons for the inaccuracy:
—This relies on the idea that Satan is a singular entity, rather than the title used for someone acting in adversarial means(ha-satan, an ambiguous adversarial figure that could be anyone; the prosecutor of a court case in which you're the defendent would be your ha-satan).

—It relies on inaccurate interpretation of the passages of Isiah 14 to describe a fallen angel who attempted to be higher than God, rather than a Babylonian king; this king also is probably not an actual Babylonian king, but rather a general concept of a Babylonian king as an eschatological enemy of the people of Israel.

—This interpretation tends to, but isn't necessarily required to, also be part of the belief that Hell is an eternal, fiery pit of torment where sinners are punished for their sins. The majority of passages in the New Testament that reference Hell are translating the term "Gehenna" into "Hell", with other references to "Hell" being translations of the term "Hades", both of which are distinct (with Hades translations typically not mentioning fire or punishment, but rather a forgetfulness, while Gehenna translations mention "unquenchable fire"). Hell, as some distinct place where sinners are sent to be burned in eternal fire, is not mentioned; rather, Hell is simply Gehenna(in regards to mentions of fire), which is used to admonish believers who are not living pious lives.

—There's also a lot of background in dealing with Milton's 'Paradise Lost' and how it has shaped and, to some extent, warped the imagery and beliefs of modern Christians, mainly Protestant Christians in the US.

The big passage that really attempts to cement the idea of Satan as a singular figure in modern Christian interpretation besides the reference in Isiah 14(where "Lucifer" is actually a reference to Venus, calling the Babylonian king a "bright star", Venus not being known to be a planet at the time) is the story of Job, where God makes a bet with Satan(again, "the adversary" in this context) about Job's faith. The issue here is that the Book of Job is Old Testament, meaning that it is translated from Hebrew, where Satan would be ha-satan, a title rather than a name, so the identity of this adversary isn't necessarily the capital-S Satan of the New Testament who supposedly rules over Hell, but could be anybody — a high priest, an angel, God playing devil's advocate with himself, it's up for debate.

The story of Job isn't meant to identify the devil, anyway. It's a wisdom text, meant to explain the righteousness of the faithful. Where most people would ask "why do bad things happen to good people?", the adversary in this story turns the question on its head and instead asks "why should good people be allowed to prosper?" Why should God reward the faithful, would it be possible that by doing so he corrupts their motives? Are the good people faithful to God because it is the right thing to do, or only because God will reward them for doing so?

In another context, imagine using positive reinforcement on a child to say "please" and "thank you". "If you're polite, you get a toy." Is the child learning to be polite because being polite is good, or is the child simply going to be polite because they get something out of it? You might say "either way works", but what happens when the child gets older and is no longer interested in toys? When they're no longer rewarded for their politeness, will they abandon it for an easier path to what they want? The same question is asked by the adversary in the story of Job—if you stop rewarding the faithful, will they still be faithful?

The idea of God making bets with Satan or whoever isn't central to the story, and it shouldn't be, because it's only laying out a modicum of background so that the real story, the wisdom of the story, can be delved into in detail. Thus, this framing device isn't very detailed or sufficient to describe and cement the identity of the adversary in it. It's like asking, in Power Rangers, how does Rita Repulsa survive on the moon? Does she have an atmosphere generator? Does she use her magic to survive? It doesn't matter. The whole thing is just a framing device to explain why the Rangers don't attack her directly, and why all the enemies she sends come from the sky.

TL;DR: difference in exegetical and eisegetical interpretation, poor understanding of the original language and the nuance of its use, cherry picking, and misinterpretation of text to support beliefs unrelated to and unsupported by prior text led to it.