if the word 'chienne' means also 'bitch' in French, this sentence may be considered like: this is my male bitch, how to say male bitch in English? Then, i've just found the appropriate word for French. :)
'chien' is the word for male dogs, and 'chienne' is the one for female dogs. Also, 'mon' refers to a masculine noun, if it was a feminine noun following it, it would be 'ma'. Hope it helps ;)