As a NES, but not a teacher:
1. "Andrew's family use transportation of different types." works for me and means the same as your example.
You can't say: "the same number of transportation". You might have said: "the same number of types of transportation", but this would be inappropriate in your case, as each family member only used one form of transportation.
2. "transportations" is incorrect, as you correctly believe.
Hope this helps
Student or Learner