Is it possible to change companion pet?

As per this question, the only difference between companion pets is aesthetics and a special effect that occurs when you feed fish to your pet.

However, I'd still like to know is it possible to change companion pets once you have started a character, or does this involve starting a new character?

In general, you will have to start a new character.

A second option is too hope for a fishdrop that let's you change your pet permanently. Those do exist, but are very rare and hard to come by.

I personally found one that changes the pet permanently into a spider.

The unofficial, fan-created editor called Torchlight 2 Rapid Respec can change the pet type for your existing character.

Using this editor, you can change your pet type by using the Aesthetics tab.

Unfortunately the perma-morph fish effects for your pet are lost if you ever accidentally rclick another fish. The modding community of Torchlight II is currently building a lot of momentum, so keep an eye on the runic games modding forums ( http://forums.runicgames.com/viewforum.php?f=46 ) and hopefully some of the clever folk there will come up with a means to change the pet model for an existing character.