"She runs like a dog."
Translation:Αυτή τρέχει σαν σκύλος.
Routledge's Greek: a comprehensive grammar of the modern language (p.400): "Σαν 'like' is used in comparative constructions with a noun phrase in the nominative or accusative.... When the noun phrase that follows it is in the nominative (1) , the noun phrase essentially denotes the same person or thing as the subject (i.e. it is a subject predicate), while a noun phrase in the accusative (2) denotes someone or something that resembles the subject:
(1) Τρώει σαν βασιλιάς (nom.) 'He eats like a king' (2) Τρώει σαν το βασιλιά (acc.) 'He eats like the king'
Example (1) means that he eats as if he himself were a king, while in the appropriate context (2) may imply that he eats in the way that a particular king eats."
I think it's in the same case as what you are comparing it with.
Here you are comparing αυτή (nominative / subject), so the thing after σαν is also nominative.
If you had been comparing objects, I think you would have needed the accusative.
e.g. Σ' αγαπώ σαν αδελφό / σαν αδελφός (I love you like a brother -- 1) like I would love a brother, 2) like a brother would love).
There was a small typo in the incubator. Αυτή τρέχει όπως σκύλος, without an article, doesn't make sense (unlike σαν, όπως in these cases can't be used without the article) so it's been corrected. Thank you for bringing this to our attention by commenting. ^.^
As for the imitation part, I don't think that the translation with the indefinite article implies imitation, or that όπως is different in usage. Σαν and όπως are, most of the times, interchangeable, with a small change in structure.
I will admit that, by using the definite article, the case becomes a bit more specific, and I could see someone thinking there's imitation. However, nothing is absolutely certain. :P