"He looks very Spanish" - is this sentence correct? Does it mean the same as "He looks like Spanish"?
"He looks very Spanish" is fine. Also: "He looks Spanish"; "He looks a little bit Spanish"; "He looks extremely Spanish"... etc. These are correct sentences.
Whether or not he is from Spain does not affect whether he can look Spanish either. You could correctly say, "I happen to know he was born in Madrid, and he looks extremely Spanish." Or, "He is from Spain, but he certainly does not look at all Spanish."
But : as @Michael Harvey and @Colin Fine say, "He looks like Spanish" is not right. A person can only "look like" a noun, (or something that functions as a noun in the sentence). "He looks like a Spaniard", "He looks like he is Spanish", or "He looks like the Spanish" [i.e. "He looks like the Spanish People look"] are OK.