Short answer:

Yes, people have been moving fruit bearing trees around the globe for a very long time. Of the three that you mention: bananas, plums and mangos-- plums were grown in continental Europe, bananas and mangoes were grown around the Mediterranean brought by the Muslims. So I doubt that these Berlin delicacies were coming from China; more likely Spain, Sicily, Cyprus or Naples, or in the case of plums, in a nearby orchard. Plenty of tropical fruits were grown in hothouses in Northern Europe as well.

Discussion:

The purpose of the transport of the full trees was not that the fruits themselves then on the trees would be useful, but rather that one could get a fruit bearing tree in the ground and start growing fruit much more quickly. Growing a fruit tree from seed means many years of growth before the tree will bear fruit, and getting seeds from wild collected trees to grow in alien soil is very much a hit or miss proposition.

Most famously Captain Bligh's breadfruits on the Bounty

The object of the voyage being now completed, we began, this afternoon, to remove the bread-fruit plants, to the number of 1015 to the ship. They were in excellent order : the roots had appeared through the bottom of the pots, and would have shot into the ground, if care had not been taken to prevent it. Besides these, we had collected many other plants, some of them bearing the finest fruit in the world, and others valuable from afford ing brilliant dyes, and for various other properties.[William Bligh, "The dangerous voyage performed by Captain Bligh: with a part of the crew of His Majesty's ship Bounty," (London 1824)]

So the purpose in transporting mature fruit trees wasn't a fruit trade, it was to generate a local fruit industry-- but there was also in fact a very healthy fruit trade, quite early. Columbus is the first European to bring back a pineapple to Europe, in 1493. The British, with their colonies in the West Indies producing sugar also produced pineapples, which were brought back across the Atlantic on the same boats.

John Evelyn the noted diarist and arborist notes of the pineapple in the late 17th century

When this fruit is grown to a likeness, you shall perceive it by the smell, which is far beyond the smell of our choicest fruits of Europe, as the taste is beyond theirs ... The rind being taken off, we lay the fruit in a dish, and cut it in slices half an inch thick; and as the knife goes in, there issues out of the pores of the fruit, a liquor, clear as Rock-water, near about six spoonfuls, which is eaten with a spoon; and as you taste it, you find it in a high degree delicious, but so mild as you can distinguish no taste at all; but when you bite a piece of the fruit, it is so violently sharp, as you would think it would fetch all the skin off your mouth; but before your tongue have made a second trial upon your palate, you shall perceive such a sweetness to follow, as perfectly to cure that vigourous sharpness.

You'll see a pineapple motif in a lot of furniture and painting of the period, and we have plenty of accounts of its availability to reasonably ordinary people of means in 18th century Europe, viz the note of the journalist and politician John Wilkes to his daughter Polly in 1768

Guildford, Aug 8 1768. My dear Polly, I set out from the kings-bench this morning at five, in a post coach and four, with the marshall and tipstaffs, accompanied by several other post chaises and coaches of our friends. We arrived here at nine. The grand jury met soon after. We were received with ringing of bells and general acclamations of the people. I have had a noble present of fruit, of which I have saved the pine apple for my dear girl. I am perfectly well. Adieu.

We also have records of people growing pineapples in Europe as early as the 1690s, so one must always inquire of exotic fruits -- were they grown locally or imported? Pineapples could be either, and you'll find all sorts of structures built to cultivate tropical fruits in the 18th century. You'll commonly encounter the term "Orangerie", as for example the museum in Paris, which was originally built by Napoleon III that his orange trees might be protected from the Northern winter. Orangeries and similar structures were common luxuries at the time, I believe Louis XIV builds the first.

The orange is originally native to China, but is brought to Europe very early-- it arrives in Spain with the Moors, with significant cultivation apparently beginning in the 10th century. Lemons arrive in Europe much earlier -- perhaps by the 2nd century CE. So we've got a very long history of exotic fruits being cultivated in Europe.

Botanical exploration of the Empire, and the return of plants and trees to the metropole was an important part of the imperial project. Men like Sir Joseph Banks are celebrated for their meticulous description of tropical flora and fauna, and return samples to be grown at establishments like Kew Gardens and the even older Le Jardin des Plantes in Montpellier, France (established by Henri IV in 1593)

Sources:

BEINART, WILLIAM, and KAREN MIDDLETON. “Plant Transfers in Historical Perspective: A Review Article.” Environment and History, vol. 10, no. 1, 2004, pp. 3–29.

DAVID, ANDREW. “Bligh's Successful Breadfruit Voyage.” RSA Journal, vol. 141, no. 5444, 1993, pp. 821–824.

Howard, Richard A. “Captain Bligh and the Breadfruit.” Scientific American, vol. 188, no. 3, 1953, pp. 88–95.

McClellan, James E. “André Michaux and French Botanical Networks at the End of the Old Regime.” Castanea, 2004, pp. 69–97.

“THE BANANA AS THE BASIS OF A NEW INDUSTRY.” Scientific American, vol. 80, no. 9, 1899, pp. 137–137.

LEVITT, RUTH. “'A NOBLE PRESENT OF FRUIT': A TRANSATLANTIC HISTORY OF PINEAPPLE CULTIVATION.” Garden History, vol. 42, no. 1, 2014, pp. 106–119.

VERDIER, THIERRY. “UNE ARCHITECTURE RÉPUBLICAINE, L'ORANGERIE DU JARDIN DES PLANTES DE MONTPELLIER.” Annales Historiques De La Révolution Française, no. 309, 1997, pp. 441–450.

Le Jardin des Plantes de Montpellier, le plus ancien jardin de France, 1593, (Association des Amis du Jardin des Plantes de Montpellier: 1983)

and -- the essential work on the ways in which the imperial powers remade the world ecology:

Crosby, Alfred W. The Columbian Exchange. Westport: Praeger Publishers, 2003