The expression is '... raining cats and dogs'. I have never heard of it raining dogs and cats.
But this is an expression, which a) is not meant to be taken literally and b) only holds when the 'cats and dogs' are taken as one (grouped) item
'... are as physically different as dogs and cats'.
In this sentence, the dogs and cats are not grouped, but are being referred to individually and separately. In addition, they are being referred to literally. In this context the phrase 'dogs and cats' is referring to two different four-legged, hairy mammals that people keep as pets.
The point the author is trying to make is that dolphins and porpoises (two very similar appearing animals) are actually quite different and that you should consider them as being as different to each other as much as you would consider dogs and cats as being different to each other. As @kojiro points out, in this context, the order of dogs and cats does not matter, yet as has been mentioned raining cats and dogs is always ordered cats, then dogs.
As an interesting aside, it seems other versions of it's raining ... for instance 'it's raining money', 'it's raining men', 'it's raining lawsuits', do refer to an abundance of the object - unlike raining cats and dogs - but the rain may be figurative. I suppose in that context - perhaps on entering an animal rescue shelter - you could say 'it's raining dogs and cats'.