My answer became outdated after the question was edited. The new answer is yes
Short answer: no.
What you are describing is a classification problem. Classification need supervised learning, meaning you dataset consists in a list of image AND a list of labels, one for each image. Labels are integer values, in you case representing breeds (a different number for each breed).
Your machine will learn to recognize breeds from you dataset. Let’s say you need 1.000 pictures for each breed to train you machine. Your trained model will fail to recognize a breed if there weren’t enough pictures in the training dataset.
If you are using deep learning, you model won’t say “this is a labrador”, it will give you a confidence value for every breed it was trained on. So, your model can express anything outside the label list (it can’t say “this is a cat”). And will fail recognizing any label with poor training.