In a traditional cryptic crossword each individual word has at least two independent things in the clue referring to it, for example, in an unrealistically simple case, "it means X" and "it's an anagram of Y", so each individual answer is very likely to be unique. In addition, alternate letters of each answer are part of another word, so the whole thing is extremely likely to have a unique solution.
However, with a combination of a bad clue and a bit of bad luck it does sometimes happen that an experienced crossword solver has to take a guess before sending in their solution. I'm not an experienced crossword solver, but I've seen them at work, and I would guess that finding a good alternative solution happens less than one time in 50. That's just a guess, though. For a better estimate one should systematically compare the solutions produced by competent crossword solvers with the official solutions.
In a crossword puzzle, each word shares letters with at least two other words, likely many more. So, without any effort from the puzzle desogner, it is very unlikely to actually have all of the words have ambiguous enough solutions that match the letter count etc. Even a crossword where half the clues are 'any word' will have a good chance of having a unique solution overall.
Of course, people don't like to play crosswords like sudoku, blindly matching letters, so there does have to be some skill to the clue design to have a relatively small amount of plausible answers, if not being entirely unique.