The short answer is no, Shakespeare was not unique in foreign settings. Marlowe's Dr. Faustus is set in Germany, for example. Kyd's Spanish Tragedy likewise was set outside of the British isles. Thomas Middleton's The Changeling involves Italians.

What might be most striking about Shakespeare is how unoriginal his plots are. While Shakespeare's wordplay is extravagant, his plots are variations on somewhat well-known stories. For example, it's believed Hamlet is based off an earlier play though Shakespeare adapted it himself. Stephen Greenblatt (an not uncontroversial name in early modern studies) suggests the play could be an outlet of grief for the playwright's son Hamnet. Source study in Shakespeare is huge; many of his historical plays find basis in Holinshed's Chronicles or the famous Mirror for Magistrates of the sixteenth century. See for example this website on sources for the King Lear myth that predate Shakespeare's play.

The foreign locales certainly offer Shakespeare a buffer between commenting on current events. I don't know quite enough about Shakespeare's alleged political motivations in setting some of his plays, but it's not a huge stretch of the imagination that he intentionally waited to publishHenry VIII until after Queen Elizabeth - Henry's daughter - died. It's worth noting that the theater was not an unchallenged medium during Shakespeare's lifetime. There was a theater ban around 1593. Though after Shakespeare's death, Puritan William Prynne wrote perhaps one of the most popular anti-theatrical tracts Histriomastix which attacked the theater as the "pomps of the devil" - despite the relative religious toleration under Queen Elizabeth, a playwright did not have full freedom of expression, regardless of where his plays were set