If the Jews were correct in their claim that Christians had come into Thessalonica claiming there to be another king, called Jesus, then the charge would be both valid and serious. Within the empire, there was no king but Caesar. Rome may appoint kings to conquered territories outside the empire proper or to client states, but Thessalonica was part of the empire.
However, Acts 17:5 ("But the Jews became jealous and recruited some worthless men loitering in the public square, formed a mob") suggests that the 'Jews' were wilfully acting out of spite, knowing that the charge was false. The reference to "worthless men" seems to suggest that these men were unconcerned with anything other than causing trouble. If the 'Jews' told these troublemakers that Jason was harbouring troublemakers who defied Caesar, I doubt that a mob like this would care much about the issues. So, we can say that the crowd neither understood nor cared about the issues that worried the Jews who incited them.
Further to the validity of the charges, and especially to Luke's intentions, is whether there is any hint of this in Paul's epistles. Raymond E. Brown says in An Introduction to the New Testament, page 268, the main objection against authorship by a companion {Luke) of Paul comes from Acts in terms of historical and theological differences/discrepancies from the Pauline letters.
There is nothing in any of Paul's undisputed epistles that shows he ever taught of Jesus as a king. True, 1 Timothy 6:15 refers to Jesus as "the King of kings, and Lord of lords," but the strong consensus of New Testament scholars is that First Timothy was written in Paul's name early in the second century Burton L Mack says in Who Wrote the New Testament, page 206, that the Letters to Timothy and Titus were written at different times, undoubtedly during the first half of the second century. So referring to Jesus as king of the Jews is out of character for Paul and belongs to a later period of time.
Writing around the turn of the century, the author of Acts might well have been unaware of whether Paul ever used this term. However, the context in Acts suggests that the purpose here was to dramatise the opposition of the Jews to the missionary work of Paul. Far from showing that Jesus' kingdom does not mean trouble for Caesar, the passage might risk the opposite. Although an unlikely scenario, a zealous official could read Acts 17:6-7 and believe that Christians really do regard Jesus as an earthly king.