The rhetoric used by Whigs/Patriots against the perceived overreach by the British government was very... Dramatic. What Samuel Adams, and many, many other politically active individuals in the thirteen colonies meant by slavery was slavery in the figurative sense. That is, that the imposed taxes and other policies by the British parliament would effectively lead to the dismantling of their natural born rights (which every Briton had). If parliament forced taxation upon those who had always autonomously tax themselves, what's not to say that they would go even further?

Yet, it is your follow-up question that is considerably more interesting. How did not only free black people but also enslaved black people react to this rhetorical use of the term slavery? Historian Jared Ross Hardesty writes in his book Unfreedom: Slavery and Dependence in Eighteenth-Century Boston that enslaved black Bostonians integrated themselves into white society. Unlike the considerably larger slavery in the southern colonies, where individual slave communities arose, enslaved (and free) black Bostonians were part of a larger, multi-ethnic community where they socialized, fought, and entered relationship with white and indigenous people with a similar background of being a 'lesser sort'. Many of them even knew how to read and write, something that was permitted by their enslavers because of religious reasons. In addition, enslaved Bostonians considered themselves to have certain rights tied to their specific spot in the social hierarchy. Free and enslaved black Bostonians used the same rhetoric used by Patriot speakers like Adams and applied it to themselves, re-conceptualizing it so as to call for the abolition of slavery.

There were four anti-slavery petitions written between 1773 and 1777 alone in Boston. All of them reflect the rhetoric used by contemporary Patriot speakers. Take this excerpt from a 1774 petition written by enslaved black Bostonians:

That Your Petitioners apprehend, they have in common with other men, a natural right to be free, and without molestation, to enjoy such property, as they may acquire by their industry, or by any other means not detrimental to their fellow men; and that no person can have any just claim to their services unless by the laws of the land they have forfeited them, or by voluntary compact become servants; neither of which is our case; but we were dragged by the cruel hand of power, some of us from our dearest connections, and others stolen from the bosoms of tender parents and brought hither to be enslaved.

Or this 1777 petition:

The petition of A Great Number of Blackes detained in a State of Slavery in the Bowels of a free & christian Country Humbly shuwith that your Petitioners Apprehend that Thay have in Common with all other men a Natural and Unaliable Right to that freedom which the Grat [Great] - Parent of the Unavese [Universe] hath Bestowed equalley on all menkind [mankind] and which they have Never forfuted [forfeited] by Any Compact or Agreement whatever

Although slavery was a rhetorical flourish used by Patriot speakers in Boston, enslaved black Bostonians took the rhetoric to heart and adapted it to suit their realities as black men and women. Although slavery would not legally end in Massachusetts until the 1780s, following the Quok Walker and Elizabeth Freeman trials, enslaved Bostonians actively used the same political strategies used by white, upper-class Bostonians, in their pursuit of emancipation. They were far from passive participants in the revolutionary era and although not all of them wrote petitions, many participated in the town's political actions.

Sources:

Unfreedom: Slavery and Dependence in Eighteenth-Century Boston by Jared Ross Hardesty (NYU Press, 2016).

Petition for freedom to Massachusetts Governor Thomas Gage, His Majesty's Council, and the House of Representatives, June 1774. (Massachusetts Historical Society)

Petition for freedom (manuscript copy) to the Massachusetts Council and the House of Representatives, [13] January 1777. (Massachusetts Historical Society)

Is this true?

Sort of.

It's popular to take the rhetoric of becoming British slaves as overly dramatic. The truth is the rhetoric was not invented by the American colonists or new in 1776. The tie between British control and being reduced to slaves was dramatically introduced in the 17th century when Governor Andros, representing King James, declared that the colonists had forfeited all their rights by leaving England and could be sold as slaves. And would be, if they continued to defy him. This was, at the time, a slight exaggeration of what the British had done elsewhere.

This quite was shocking at the time and echoed down to the Revolutionary colonists.

Unsurprisingly, Andros was not very popular. But his administration, and the English Civil War, were the closest analogy the Revolutionary era colonists had to their situation. In fact, the English Civil War and Glorious Revolution both had direct influence on the American Revolution. So it's not so strange they'd reach back to something Andros had said. It was no so distant either. At the time of the Revolution, Governor Andros had been overthrown only 85 years ago. To put that in perspective, that's roughly today to the New Deal and Hitler.

The 18th century British weren't planning to enslave Americans. But their governors had threatened to in the past. They actually had indentured people, which could mean working on a slave plantation. And they had done so in the context of disputes about colonial rights.

On top of that, the British were espousing a version of Andros' absolutist argument. They put it more diplomatically, but still. Andros' legal argument was effectively that Americans had had rights as Englishmen. But in removing themselves from England they had ceased to be Englishmen and so had no rights of any sort. Notably, that their colonies and charters were at the pleasure of the King. He specifically claimed he could even enslave them, as the right to not be enslaved was in Parliamentary legislation that applied to Englishmen.

This was similar to Parliament's position in the Revolutionary War, except they vested power in themselves rather than absolutist King. Under those circumstances, the claim didn't look particularly absurd.

More than that, it proved a point. Unlike Andros, Parliament was not saying it wanted to sell the Americans into slavery. It just wanted to be able to ignore their local governments and rewrite their charters and pass laws they were bound to obey. But if Parliament could do that, Americans had no rights Parliament was bound to respect. How was it different than King James and Andros? How is having no rights different from being a slave? Are you really going to trust Parliament when they say they want to take away all your rights, the guy who threatened to enslave you was right, but... you know they totally won't enslave you. (Non-binding) Pinky promise.

The statement was also rich metaphorically. The British relationship with the colonies was exploitative. But it had resonance beyond just comparing the colonists to the African Americans metaphorically. Indeed, the reason enslaving Americans was off the table in 1774 was only because of rising anti-slavery sentiment in general. Britain would ban the slave trade about three decades later and repeatedly softened indenture and transportation. The British stopped transporting Americans about the same time they stopped transported other imperially peripheral groups.

What kind of slavery was he thinking of, and did it differ from chattel slavery of African people?

Well, slavery in Andros' time was a relatively new institution. He took up the governorship of the Dominion of New England in 1686. There hadn't been slavery as such prior to about 1660. There had been black people who had been transported from Africa, bought at the market, and put to work. But they were not yet a legally distinct group. Slavery changed significantly from about 1660-1690. Prior to that, Africans were largely put into a common class with whites who had also been transported from Europe, bough at market, and put to work. However, this system meant eventual freedom for both whites and blacks. While it's definitely the predecessor to slavery and a vastly evil institution, it was not slavery.

That said, Andros used the word slave specifically. It was meant to make a dramatic point about the colonist's lack of rights. They were not going to be transported or forced into indenture or put to labor. They were going to be reduced to slavery as were their descendants. It was also meant to underline his argument that they had stopped being Englishmen, since slaves were always foreigners or the descendants of foreigners.

That said, it's doubtful he was going to literally put them into the same category as black slaves. However, the 17th century British had captured entire populations, shipped them to places that needed labor, and sold them to people who needed labor. These people were often the same people who owned slaves.

In short, Andros said he was going to enslave them and that meant slavery like we think of it. This had never been done before and was not likely to happen. Though who knows with Andros and the 17th century English. However, it was an entirely realistic scenario (one that had happened to 17th century Americans) to be taken by the government, manacled, and shipped to labor on a sugar plantation.

From The Glorious Revolution in America, The Imperial Executive in America, The American Revolution, Origins of the American Revolution, and A History of American Political Thought.