The 12 Imams are not exactly like the Prophet Muhammad in that each of them were separate individuals and had his own life, experiences, personality, features, etc. (That is, they did not share the same soul.)
However, they all shared the same teachings of the Prophet (S). This is the job of the Imams (A) is to correctly explain the teachings of the Prophet (S). According to dominant Twelver Shi'i belief, knowledge of the correct Prophetic teachings is a divine gift transferred to one Imam after the death of the previous Imam; therefore, they do not make mistakes in the teachings or disagree about what the Prophet (S) taught.
Also, the Prophet and 12 Imams are all considered sinless. They all obviously also shared a strong commitment to Islam.
Maybe this is one reason that they tend to appear the same - it is our flaws and faults that oftentimes distinguish us the most from each other!
The rational reason for this is that Allah appointed them to preserve and expand upon the message as was necessary. If you think about it, most Muslims came to Islam in the latter part of the Prophet's life. Many of them really didn't have enough time to learn much about the religion at all, especially because the Muslims were busy fending off attackers and dealing with a lot of practical things. And after the Prophet, not all the Companions agreed on everything. So it makes sense that if Allah is going to appoint a prophet with the final message that Allah would be sure that message is protected; this is done through the imamate.