Only 1 is true because if you take an example of 3 vertices with weights as 1,2 &3 ..
A---1------B
B------2--------C
C--------3---------A
1. Minimum spanning tree would weigh 1+2=3 (only one mst possible with AB & BC)
2. whereas it wouldn't have a unique shortest path between two of them.
1+2=3 possible (AB +BC) & only 3 possible (CA) ~ two possibilities
Correct me if I am wrong!