1. He wrote how poets behave when they come for an opinion of their poetry. The assumption here is that he's implying that poets come to him for his advice on their poetry, further implying that his reputation with poetry is well-regarded.
2. He called himself a poet. Further than that, he called him self a "concealed poet". He called himself this to another poet--John Davies, a distinguished poet who was on his way to meet James I to assume the English throne. The implication here is that Bacon wouldn't have identified himself as a poet to a known distinguished poet unless his own reputation, even as an anonymous poet, was well-regarded. It also shows that his anonymity wasn't complete, that others, such as Davies must have known of his concealed poet status. Also, this letter from Bacon to Davies was in the hand of one of Bacon's secretaries, though it was signed by Bacon in his own hand. This shows further that Bacon's anonymous writings were known by some others who honored his desire for anonymity.
3. Bacon did write some poems, some to his name, and some other that are argued to be his though not in his name. He wrote some Psalm paraphrases under his name in late 1624. These are not on the same quality as the Shake-speare sonnets or early poems. However, at this time an explanation for their poorer quality (assuming Bacon did write the Shake-speare poetry) is that by late 1624 "Bacon was an old and broken man, undergoing a prolonged sickness, perhaps with declining powers, and within 18 months of death." And he probably hadn't written any poetry since 1613 when he was appointed Attorney-general, so that his inspirational powers were weaker. Further, even trying to paraphrase the psalms into verse was probably a poor judgment on his part as they don't lend themselves well to this change without subtracting from their effect.
4. In 1599 he says he wrote a sonnet to the Queen who was to dine with him. This was after the Earl of Essex made his disastrous return from Ireland. And it's been surmised that possibly this sonnet was an effort to lessen the damage to Essex from the Queen's anger. And that possibly this sonnet or it's theme found it's way into the play The Merchant of Venice where Portia gives her speech to Shylock "The quality of mercy is not strained, it droppeth as the gentle rain from heaven, etc." It does seem odd that Shake-Speare in this speech of Portia speaks of the Mercy of the monarch as above the power he/she wields. What does this aspect of a monarch have to do with a moneylender? Wouldn't a somewhat despised moneylender relate better to, say, a Duke than to a monarch? In any case, we don't have the sonnet that Bacon wrote for the Queen. But the assumption is that based on Bacon's language skills and poetic reputation that his sonnet would likely be of good quality if he felt confident enough to read it to the Queen.
More importantly is whether Bacon's style in these psalm paraphrases is like Shake-speare's style. Cockburn says they are similar in diction and the use of alliteration. And there are a number of verbal parallels in his paraphrases with phrases in Shake-speare. For example, both Bacon and Shake-speare use the term "line of life" (Bacon regarded life as a line that moves forward). "Mounted high" is used by both; the both use "frail mortality"; they both use "thou to them" or some other version having 2 personal pronouns joined by a preposition. These are just a few of many similarities.