There are several variants of the song Je cherche après Titine (probably short for a woman called Martine or Christine). The line you quote does not appear in the original lyrics written in 1917 where Titine is definitely a woman. It’s this first version of the song, that the American soldiers who had been fighting in France at the end of WW1 brought back with them to the US where it became very famous as a symbol of France, and Paris, at the end of WW 1 and early twenties. An article in French about Gaby Montbreuse, who first sung the song on stage.

A remake of the song was made in the twenties with different lyrics, and these new lyrics develop an ambiguity that allows us to think Titine is a dog.
At one point the narrator describes Titine physically with the words:

Elle était frisée comme un ange
Et s’tortillait tout en marchant¹

which could be applied to a dog, more precisely to a poodle (frisée)

and to her character with:

Titine avec son coeur frivole
Changeait de fleurte dix foix par jour²

which can refer to a dog as well as to a fickle unfaithful woman.

Thus in the concluding line:

Une chienne qui a vraiment du chien

chienne can be understood with the two meanings of the word, just as the word "bitch" in English: the animal and the derogatory term for a woman.
The line further plays on words by the use of the turn of phrase avoir du chien which contains the word chien without refering to a dog.

^{1 "she was curly like an angel and wiggled her butt along"
2"Titine’s heart was frivolous and kept flirting around all day."
(Translation mine, not literal).}