Due to the long lifespans and long gestation periods of elephants, breeding them as one does with horses was impractical at best. There are no known acknowledged "breeds" of elephants (aside from the Asian and African varieties that we know of today).

Since /u/BuddHasLittleWarlock has already pointed out the problems with breaking certain traits in elephants. I just want to add that there really wasn't a need for the Mughals to breed selective types of Elephants. The current type of elephant in India already did exactly what the Mughals needed them to do.

The Mughals used elephants as:

A) pack animals

B) platforms for their generals to observe the battlefield and give commands

The size and power of the Indian elephant was already perfectly suited to these tasks so they really had no reason to breed smaller ones or even bigger ones.

As a fun little fact, armies would often use marksmen to pick off the generals on the elephants, or they would target the elephants with artillery sending them panicking through their own lines.

It is very difficult to breed Elephants in captivity, even using modern methods such as artificial insemination, as captivity significantly reduces fertility in both males and females.

Even today with modern science it's considered a BIG deal if a zoo can produce a healthy calf from captive elephants.

Pre-modern societies that used Elephants would not have been able to selectively breed them, there may have been cases where captive births happened, but they would not have been able to systematically control this as they would have been able to with dogs or horses. Instead they would have relied on replenishing their stock from the wild.

http://nationalzoo.si.edu/SCBI/ReproductiveScience/ElephantBreedRepro/

While this doesn't answer the question, I would like to add that the Mughals did not use Elephants like Westerners used horses. In fact one of the reasons that the Mughals were successful against the ruling elite of Northern India was that the Mughal's battle tactics were able to frighten the elephants used by them.

Also, the mughal armies had Sowar (सवार) or horse mounted knights just like western ones.

Source: Medieval India: A Textbook of History for Middle Schools Book by Romila Thapar