Shakespearian? Far from it.
Apart from the fact that shakespeare wrote in Iambic pentameter, his character development was deep and philosophical in his tragedies and blunt yet subtle in his comedies. The myriad facets of each character, the details given are such that no videogame to date has equaled them.
To compare Chrono Trigger's story to Shakespeare is blasphemy in literary circles. Much of the detail attributed to the characters and story has been given in this very community. Most of which is speculation.
When you get down to the nitty-gritty, the story in chrono trigger isnt all that deep unless you extrapolate from it.
Many of shakespeare's stories are shallow too, but are made up for by the many shallow details. But conversely, many are very deep. Hamlet is one example.
You simply cannot compare the two. Aside from the fact that they are completely different writing styles, Shakespeare was arguably the best writer ever, and the team of writers on CT were prbably far from that.