Is dolphin both singular and plural?

Yes; however, the preferred plural of dolphin is simply dolphins. In fact, most dictionaries I have viewed-Merriam Webster's included-do not even list dolphin as an acceptable plural form of the marine mammal. Thus, dolphin as a plural form can be considered substandard.

What it boils down to is this: Unless you strongly prefer using dolphin as both a singular and plural noun, use dolphins when speaking about more than one dolphin. However, when someone else uses dolphin as a plural noun, realize that he or she is not incorrect.