Yes; they are a different part of speech.
σαν is a preposition, so it stands before a noun or noun phrase ("like a bird", "like you"), while όπως is a conjunction, so it stands before a clause that can include a verb ("as you know", "as she told me").
Αυτή τρέχει σαν σκύλος include a verb ''τρέχει''. Αυτή τρέχει όπως ένα σκύλος it is also correct? Κίτρινο όπως ένα λεμόνι doesn't include any verb ... So Κίτρινο σαν ένα λεμόνι it's correct also? ...or is it something i still don't understand!
I don't think that Αυτή τρέχει όπως ένα σκύλος is correct because there is no verb.
Αυτή τρέχει όπως τρέχει ένας σκύλος could work, though.
My greek friends just told me that this is wrong. You can say Αυτή τρέχει όπως ένα σκύλος. And your second sentence sounds weird.
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.
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
For some reason, the Ancient Greek conjunction from where it comes, ὡσάν ('as if it were') usually omits the article too when it refers to person, so probably the usage has extended in Modern Greek.
Is that a real Greek expression? If so, is it good or bad? In English it is bad.