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.