Plum, in this case, does not refer to the fruit, but is rather an adjective meaning "good"... maybe. Plum may also be a reference to sugarplums, which was simply a sweet sort of candy, or to the raisins that were often used to sweeten the pudding, and were sometimes referred to as plums.
A plum pudding is a sponge pudding made with fruit and brandy and sweetened with sugar, although the details have varied greatly over the centuries. Early recipes used beer and beef; sometimes the sugar is mixed in, sometimes poured on top; and they were traditionally boiled, but the Victorians preferred them steamed. They have always been highly spiced.
"Boiled plum or Christmas pudding: Cut a pound of beef-suet extremely fine, to which add a pound of raisins well stoned, half a pound of currants, picked, cleaned, and dried, some nutmeg, two spoonsful of brandy, two ounces of candied lemon-peel, and one ounce of candied orange-peel shred fine, six well beaten eggs, a gill of cream, and seven or eight table-spoonsful of flour, mix them well, and boil it four hours; when done, serve with melted butter and grated sugar."
--Modern Domestic Cookery by Elizabeth Hammond (1819), Sourced from Foods Of England.
It is now most common to refer to plum pudding as Christmas pudding, and they may be treated as synonyms; technically, they are different in that Christmas puddings are eaten at Christmas, and plum puddings may be eaten at any time of year. Granted, that that year is statistically prior to 1900, but you could have a summer plum pudding, if you wanted. Plum pudding was also known (more accurately) as Currant pudding, and is essentially an over-sweetened, milk-and-egg-rich form of a fairly standard type of pudding which had dozens of names depending on subtle variations; Hunter's Pudding was the same thing without added sugar and with as little milk as possible, allowing it to store much longer; Shelford Pudding was Current (or Raisin) pudding without added sugar.