Generalising is not the problem. It is most of what learning is: you touch a hot stove once and conclude something reliable about stoves. The fallacy lies in the mismatch between how much evidence you have and how much weight you put on it — a conclusion about thousands drawn from two.
Sample size is the obvious failure, but the subtler one is representativeness. A large sample drawn badly is worse than a small one drawn well, because size lends false confidence. Survey only people who answered your call, and you have learned about people who answer calls. This is why survivorship bias fools so many people: studying only successful companies tells you nothing until you check whether the failures did the same things.
The everyday form is the anecdote. A remedy worked for someone, so it works; a person had a bad experience with an airline, so the airline is terrible. Anecdotes are vivid, personal, and memorable, which makes them psychologically far heavier than the statistics that should outweigh them. They are not worthless — a single case can disprove a universal claim, and can point toward something worth investigating — but a case is a lead, not a conclusion.
The most damaging form is stereotyping: observing traits in a few members of a group and extending them to all. Logically it is the same error as any other hasty generalisation, but the costs are not symmetrical, and it is reinforced by confirmation bias, which makes fitting cases memorable and exceptions forgettable. Naming the structure matters here precisely because the reasoning feels like experience rather than inference.