That doesn’t follow. Solving it only proves that it has a solution. It says nothing about whether there’s another solution.
The easiest way a sudoku can have two solutions is where there’s four cells with these properties:
- are on the corners of a rectangle
- all four are initially blank
- there’s only two different numbers in those four cells
- the four cells are contained in two 3x3 blocks
In those cases, there are two solutions with the four cells having these two solutions:
As I understand it, most sudoku generators go through a check stage where they look for this situation and eliminate any puzzles that have it.
However, you can have more elaborate double solutions involving 6, 8, 10, etc. cells in half that many blocks. At the extreme, it involves all 9 blocks, which means that there would be two numbers that are not in the initial puzzle. This last condition is easily checked, and I expect the generators eliminate those as well. I don’t know if they check for the intermediate double-solution conditions as well. I suspect they don’t.