As to the oddity of the story, first of all, this is not in Gemara or even Medrash Rabbah. So, if you have a hard time with it ignore it. There are other weird stuff about Shlomo Hamelech out there which, given the obsession of other nations about Shlomo Hamelech, I assume there are stories that originate from without.
This is what happened with the famous story of the two brothers trying to lighten the load of the other in the middle of the night, on the spot of the future Bayt al-Maqdis. It appears to be an Arabic story that made its way in a couple of centuries ago.
However, being that Tosafos mentions it, I’ll regard it as legitimate.
Now, a Sheid is not completely a physical being. They drive physical things and are basically related to the material world, but they aren’t actually composed of matter. Perhaps it can be said that just like our soul is manifest in a body, their personality is manifest in motion, changes, and action. Therefore, the whole story about the capture of Ashmedai should be read in this light. Shlomo Hamelech knew how to capture this non-physical being to be able to utilize it’s resources. To the outside observer, there was nothing to see other than amazing success.
In this story too, this strange guy showed up from nowhere, or perhaps he was born to a local family this way (which would explain the marrying and application of Jewish law) and the Medrash is informing us that this was Ashmedai’s doing. He pulled up this form of human from other worlds and applied it here.