The sentence in concern is : We agreed that Coco and Rocky are not the same dog. Shouldn't it be: We agreed that Coco and Rocky is not the same dog. ? According to my Word grammar editor, is is the correct form of the verb there. It doesn't sound right, though.
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The fundamental rule (in US English, at least) is: if you have two subjects connected with "and", then you must take a plural verb. By comparison, if you have two subjects connected with "or," then the verb takes the singular/plural to match the second subject. "The Smiths or Mike is going to do the job." and "Mike or the Smiths are going to do the job."
As the comments suggested, regardless of how many actual items the subject nouns refer to, the fact that you have multiple subjects means the verb form must be plural.
PS trusting MS Word on grammar or spelling is even dumber than trusting MS Excel on math.
The verb changes only due to the subject: in this case two dogs, therefore a plural verb. This is irrespective of the fact that the object is singular.
And although Word's grammar check can be useful, don't take is as gospel.