By the most part, yes, but let me explain why is not fully yes. Most of Shakespheres works are actually written iambic pentameter, a type of verse in poetry, also happening to be one of the most popular.

That being said, people at the time did not speak generally in that "perfect" speech, espaiclly lower classes. It may have been used in speeches given by upper class and royalty. Many of idioms and words used in his plays were used, just not the style that his plays are written in.

Also note, you are speaking the same type of English as in those plays, as Shakeperean English is modern English.

Shakespeare also varied the way his characters spoke. He uses the iambic pentameter in the speeches of kings and lords, while lower-class characters are more prosaic. Magical characters (Puck, Oberon, and the fairies in A Midsummer Nights Dream; The Witches in Macbeth; Ariel in The Tempest) often had non-iambic but still poetic meters. Poetic or heightened speech often communicates the nature/background/status of the character. Kings will just rhyme lines sometimes ("High-stomach'd are they both, and full of ire, In rage deaf as the sea, hasty as fire." - King Richard II, I.i) as a way to show their status and sound fancy compared to others.

Another example is in Act 1 Scene 5 of Romeo and Juliet. When R&J first meet, their lines sync up in meter and rhyme scheme so that their first interaction forms a sonnet, a popular form of poetry at the time.

People didn't go around casually spitting sonnets at the time, but here it communicates their status as uncommonly bound lovers to the audience. It's as if you started flirting with someone and the resulting conversation turned into solid freestyle rap.

Source: Am a theatre major, many a lecture on the old Bill "the Bard" Shakespeare.

In conjunction with the historical perspective, it's useful here to relate this to what you're talking about - artistic license and/or hackery. Shakespeare's characters - much like normal people do - use different levels of speech so that the audience can perceive different personalities, different states of mind, different levels of status, etc.

As has been stated elsewhere here, people of high status will often speak with much formality. You'll also often see moments of "poeticism" with characters who are mocking each other, or deliberately obfuscating, or trying to seduce one another.

Take a look at Hamlet, Act 4 Scene 2 and at Hamlet, Act 4 Scene 3 as examples. Hamlet is being deliberately obtuse and wasting everyone's time by talking like a teenager with a thesaurus who belongs in /r/iamverysmart. The people around him are speaking very clearly - "Where is Polonius?" and "What have you done, my lord, with the dead body?" - and he is responding with statements like "Compounded it with dust, whereto 'tis kin" and "Not where he eats, but where he is eaten: a certain convocation of politic worms are e'en at him." In the following quote he gives up the game (in response to "Where is Polonius?"):

"In heaven; send hither to see: if your messenger find him not there, seek him i' the other place yourself. But indeed, if you find him not within this month, you shall nose him as you go up the stairs into the lobby."

In most productions, "as you go up the stairs into the lobby" gets a laugh, because it feels shockingly direct and modern in response to all the obfuscatory near-nonsense he's been spouting since he decided to feign madness several acts ago.

Here is an especially good example of Shakespeare using this type of contrast. It's from Merchant of Venice, acted by Ian McKellan and David Suchet as part of a BBC program on how to approach Shakespeare as an actor. Except for a few outdated wordings, Antonio's Speech is easily understood and direct - "In sooth, I know not why I am so sad / It wearies me; you say it wearies you". By contrast, the speech that follows is dense with wordplay and ornament, and the actors and director go into the reasons for that particular character to be using heightened language - mockery, playfulness, showing off, etc.

Anyway, you are absolutely right to compare Shakespeare's non-naturalistic language to modern non-naturalistic language. We take it for granted when an Aaron Sorkin character is able to rapid-fire through several acronyms and statistics, quote Latin, and speak in large, fast paragraphs, or when Miranda Cosgrove uses an especially roundabout sentence construction like "Sexy is what I try to get them to see me as after I win them over with my personality." In Shakespeare, because the conventions aren't as familiar to us, we dismiss them as poetic, high-falutin, or antiquated even though the dramatic device of "Powerful person using heightened language among normal-speaking peasants" is usually the same whether you're watching Thor, The West Wing, or a Shakespeare History.

Finally, it's worth noting that people do this in real life. Not everyone speaks in what we consider "naturalistic" speech all the time. This slightly-suspect study notes that men tend to use words of lesser frequency (aka fancier words) when primed for 'romantic' conversation. People frequently use heightened language in order to be funny, or to appear smart, or to be playful. Formal speeches follow their own conventions. And long-winded blowhards often write entire essays that are nearly incomprehensible gibberish. Shakespeare's plays tend to have a robust mixture of linguistic styles, and many excerpts have a lot in common with modern speech, which indicates that your suspicion might be correct.

Edit: I a word