I'm pretty sure we should get rid of the period in both sentences, as this appears to be a sentence fragment. The lack of any kind of verb threw me off here, since usually you'd only omit it if it were van in a construction like "ez a második autó a kék háztól".
Yes, this is what can be called an incomplete sentence. I think it can still be a sentence though. It could be an answer to the question: "Which car is yours?".
But I see how it can throw somebody off. To make it more pronounced, I can insert an "egy":
"A kék háztól a második egy autó". - "The second (object) from the house is a car".
It can be an answer to a question, a statement, but I'm not sure whether I'd call it (or "Yes." or "Sure." or "The red one." or other possible answers to questions) a "sentence".
OK. Maybe it is a difference in terminology. But, I think, Hungarian grammar looks at it this way: (almost) everything is a sentence, but some sentences are incomplete. Something is missing from them but they are still sentences. So we don't have sentences and "whatshouldicallems", but instead we have sentences, some of which are incomplete.
Agreed -- this would be more recognizable as a fragment if there were no period (or upper case letter at the beginning).