English Translation
My love for my dog was very deep.
浅からぬ means “not so shallow”, and thereby it means “somehow deep”. But 浅からぬ doesn’t mean “VERY deep”. 浅からぬ is equivalent to 浅くない (adjective 浅い + negation ない). ぬ is an old-fashioned way of expression and still frequently used in modern writings (especially with some poetic nuance).
if you say “I do NOT say I dislike it” in English, you mean that you are okay with it. But the English expression in negation doesn’t mean that you like it VERY much. That’s the same as 浅からぬ (“not so shallow” is not equal to “very deep”).