Sadly we don't have textual attestation to Shakespeare's earring (not even a funny anecdote from Ben Jonson) or material proof of it, like a reference to it in Shakespeare's will. Shakespeare is depicted without an earring in the engraving accompanying the First Folio -- the only depiction of Shakespeare that is unambiguously intended to be Shakespeare -- but that absence doesn't necessarily mean that Shakespeare never wore an earring, either. Earrings were a fairly new, fairly trendy presence in women's jewelry at the end of the 16th century. (They had classical and medieval precedent, but were not continuously considered broadly fashionable in Europe.) Their appearance in men's fashion was likewise novel enough to raise eyebrows with people who were already negatively disposed toward unnecessary adornments for both men and women. But as far as fashionable Elizabethan men were concerned, the earring was one of several ways to showcase personal style and luxury -- alongside rings, chains, aglets and buttons, decorative belts, and so on. Fashionable Elizabethan men weren't afraid of jewelry, when their social class and income allowed for it.

In his polemical pamphlet The Anatomie of Abuses, Philip Stubbes files earring-wearers alongside other specimens of shameless sartorial excess, describing the practice as some kind of barbarian self-mutilation. His primary topic in this chapter is women's fashion, but not only women:

Another sort of dissolute minions and wanton Sempronians (for I can term them no better) are so far bewitched as they are not ashamed to make holes in their ears, whereat they hang rings, and other jewels of gold and precious stones. But what this signifieth in them, I will hold my peace, for the thing itself speaketh sufficiently. There is a certain kind of people in the Oriental part of the world that are such Philauto lovers of themselves and so proud withal, that having plenty of precious stones and margarits [pearls] amongst them, they cut and lance their skins and flesh, setting therein precious stones, to the end that they may glister and shine to the eye. So, except these women were minded to tread their paths and follow their direful ways in this cursed kind of unheard-of pride, I wonder what they mean. But because this is not so much frequented amongst women as men, I will say no more thereof, until further occasion be offered. [emphasis mine]

Stubbes is trafficking in moral outrage here, but you can get a sense of the qualities attributed to the earring and its owner across lines of sex -- novelty, exoticism, wealth, pride in personal appearance. This is all stuff that, in less pessimistic terms, adds up to earring-wearing as quite fashionable. The earring depicted in the Chandos portrait of William Shakespeare seems to be a pretty typical Elizabethan men's earring, even on the modest side with no precious stones and margarits in sight -- you can compare it to a Nicholas Hilliard portrait of an unknown man wearing a plain gold hoop interlocking with another gold hoop. Sir Walter Raleigh is depicted wearing a more elaborate pearl earring here in one 1580s portrait, but later portraits depict him without such conspicuous jewelry. (Francis Drake is also depicted with a pearl earring.) Shakespeare's patron Henry Wriothesley was depicted in one 1590s portrait wearing a type of earring that doesn't seem to fall neatly into either the gold hoop or pearl drop category, and Shakespeare wouldn't be the only poet pictured with an earring, either -- a young John Donne circa 1591 sports an earring with a cross-shaped dangle. At any rate, some Elizabethan men wore earrings of different styles, and some of them may have left off their earring-wearing in later life. These men are courtiers and adventurers as well as dashing youths -- the tie to supposed nautical traditions of ear-piercing might explain Drake and Raleigh, but fashion alone would do plenty to justify why all of these men chose to pierce their ears and to be depicted wearing earrings. Notably, all these men are depicted with single earrings, but this may be an artifact of the ¾ pose they're depicted in -- a Hilliard miniature of a young woman (painted roughly contemporary to the louche-looking man) shows two earrings, both gold rings with pearl drops, but other portraits of high-profile women like Queen Elizabeth I are painted in similar poses showcasing a single earring. A few Elizabethan/Jacobean writers attributed an undesirable ambisexuality to earring-wearing -- treating it like a mild form of cross-dressing, and a parallel to other developments in the fashion world that were seen to blur the boundaries of male and female garb, like men having long frizzled hair and women wearing mannish doublets. A lot of modern writers have likewise projected sexual ambivalence on the Elizabethan male earring, at least when it comes to Wriothesley and Shakespeare -- I've never read anything attributing any particular sexuality or sexual ambiguity to Raleigh or Drake on the basis of their earrings alone, but maybe it's out there. Some of the 19th and 20th century pushback to the idea that the Chandos portrait depicts Shakespeare is also flavored with a disapproving tinge of sexuality -- if you like the portrait, it makes Shakespeare look rakish and modern, even sexy, but if you dislike it, it makes him look effeminate or un-English.

Is Shakespeare depicted with an earring as a nod to his identity as a poet? Some Early Modern English poets wore earrings, as evidenced by Donne, but other literary contemporaries of Shakespeare like Ben Jonson were not depicted with or otherwise reputed to wear earrings, at least not as far as we know. (Our only portrait of Jonson comes from later in his life in the 1610s, so perhaps he had one as a young rakehell, it's hard to say.) A single earring wouldn't be the exclusive territory of poets, but the earring seems to have been associated with a certain strain of conspicuous, perhaps dashing masculinity. The impression is cultured luxury, a little rakish, a little dashing -- but some of that impression in the study of the Elizabethan male earring may be drawn from these high profile individuals having earrings, rather than being the rakishness of earrings explaining their popularity among such individuals. (Indeed, Shakespeare being the most high-profile Elizabethan poet and the Chandos portrait being the most high-profile image of Shakespeare would be enough on its own to cement the association in the popular imagination whether or not other men wore them.) So Shakespeare might have had an earring, and the man in the Chandos portrait might have been Shakespeare. I personally don't doubt that the Chandos portrait depicts Shakespeare, and it's well within the bounds of possibility that Shakespeare wore an earring. But we have so few images or physical descriptions of Shakespeare to draw from that we can't say conclusively that Shakespeare wore an earring, or when in his life he wore it, or why he wore it.