The indefinite article is actually omitted because the meaning of this sentence is a general one. She runs like a dog would in general, like any dog would. For more info about the article omission, check the Indefinite Article Omission that's under the tips and notes here https://www.duolingo.com/skill/el/Basics-2
As a native speaker, I must say I agree with Mizinamo.
Όπως is translated to as, like, such as or however, but it is more commnly used with a verb.
Κάν'το όπως θέλεις. - Do it however you like.
Όπως...; - Such as?
Όπως βλέπεις, είναι πολύ μικρό - As you see, it's very small.
Even when it is not used with a verb, it is usually followed by something that makes the subject distinct rather than general (like the one with ένας σκύλος, where it could be any dog).
Κάν'το όπως αυτός - Do it like him.
Κάν'το όπως η δασκάλα - Do it like the teacher.
Even in this case
Don't behave the way he does - Μην συμπεριφέρεσαι όπως συμπεριφέρεται αυτός.
Could also the difference be in that σαν compares the quality of the entities being compared, while όπως compares the quantity?
I mean, in the example Κίτρινο όπως ένα λεμονη you might mean that something is yellow but to a degree (quantity) similar to that of a lemon. Or maybe Το αγόρι είναι κόκκινο όπως μία ντομάτα to mean that the boy is almost (quantity) as red as a tomato (after he ran or he is embarrassed, maybe).
Meanwhile, I could say Αυτή τρέχει σαν ένα άλογο/μία χελώνα to mean that she runs like a horse (fast) or like a turtle (slow), indicating quality.
Perhaps even both? Αυτός τρώει όπως/σαν ένας σκύλος to mean that he eats as much as a dog eats (quantity) or in the way a dog eats (quality).
I really would like to hear some feedback haha since in Spanish we can make those kinds of comparisons but with the same preposition, but I don't know if it's the same in Greek