Sorry, neither is correct. You can't modify the 'so do I' idiom. You can only use it verbatim as a response to something that someone says. I love dogs, (and) I love cats too.

Sorry, but all sentences you've suggested are fundamentally wrong: you are trying to do something impossible with the grammar of the language. 'So do I' means '.. and me, too': it can never mean 'and cats too'. Please accept that however hard you try to force 'So do I' into a sentence starting "I love dogs", it will always be wrong. The purpose of phrases '"So do I", 'So have I", So can I" and so on is to emphasise a new grammatical subject of sentence: in this case, 'I'. We use them to describe a situation where the verb and complement are the same as in the original statement but where the subject is different. We start with 'So', 'Neither' or 'Nor', add an auxiliary verb ( to avoid the repetition of the verb and its object) and then add a different subject from the original sentence. For example: "Jack loves dogs, and so do I." This means "Jack loves dogs, and I love dogs, too." As you can see, the first and second parts of this sentence have the same verb (love) and the same object (dogs). The additional information provided by the phrase "so do I" is a new SUBJECT. Your mistake is to try to use "So do I" (which is all about new subject, 'I', as opposed to the original subject, for example, 'Jack' ) in order to introduce a new object ('cats', as opposed to the original object, 'dogs'). This just cannot be done. The only way you can use a 'So + auxiliary' phrase for the situation you're trying to describe is if cats and dogs are the grammatical SUBJECT of the sentence. For example: "Dogs are a favourite animal of mine, and so are cats." "Dogs have always been special to me, and so have cats." Can you see that the consistent element in this sentence is speaker and the idea of loving an animal, and the new element in the second half is the additional subject - cats. If you look at you examples from COCA, you'll see they follow this pattern - the same complement but a new subject. Remember : 'So do I" means 'Me too'. I hope that's clearer now.

"I love dogs and cats" This is more natural. We say "So do I" if someone else likes dogs/cats. Girl: "I like dogs" Boy: "So do I!"

Thanks, Frazier. I totally agree that I need to keep such sentences parallel in structure. So, another way of saying it could be 'I love dogs, and so do I love cats". Here, I keep the sentence parallel in structure by using love in both parts. Does it sound better? Perhaps, it will sound better if I add more to it. I love dogs very much, and so do I love cats. What do you think about *last variant?

The second is better, but I wouldn't say it's right. You need to keep your sentences parallel in structure if you omit words. I do love dogs, and so do I cats. It's still extremely awkward. I'll give you some natural alternatives: I love dogs like I love cats. I love dogs as much as cats. I love dogs as much as I love cats. I love dogs as I do cats. (That last one implies, though doesn't expressly mean, that you treat or care for them in the same way.)