An MST is unique if all edge weights are distinct. If edges have the same weight, you might have multiple MSTs with the same total cost. For example, if two edges have weight and you only need one of them, you can pick either.
Both choices give you a valid MST with the same total weight. To check uniqueness, look for edges with equal weights. If any exist, the MST might not be unique. You would need to check all equal-weight edges.