Complexity of vector comparison
Normally, in asymptotic analysis, it's most typical to analyze the worst-case running time.  It is correct to say that the worst-case running time of comparing two $N$-vectors is $\Theta(N)$; and not any less.

However there's nothing saying you have to focus on worst-case running time.  For instance, perhaps you are interested in the average-case running time (also known as expected running time).  In this case the answer will be $\Theta(1)$ if we assume that the vectors are uniformly distributed, as you predicted.

However average-case running time is tricky.  The expected running time depends on the distribution of values, so you must specify a distribution before you can begin the analysis.  If the distribution isn't uniform, then the expected running time might be very different.

Also the expected-running time doesn't add up like you might think.  What happens if you run Quicksort on a list of vectors?  Can we say that Quicksort does $\Theta(n \log n)$ comparisons on average, and each comparison takes $\Theta(1)$ expected time, so Quicksort on a list of vectors will take $\Theta(n \log n)$ time?  No, that doesn't follow.  Quicksort will tend to move similar vectors closer to each other, so as you get to later stages in the algorithm, it's possible that you might be mostly comparing vectors that are similar to each other.  In particular, the two vectors you're comparing won't be uniformly distributed; they'll come from some joint distribution which tends to make similar vectors more likely.  Of course similar vectors are (on average) slower to compare than independently uniformly-random vectors, so there's no longer any guarantee that every comparison takes $\Theta(1)$ time on average.

So, average-case running time analysis can be a lot trickier to get right than worst-case running time.  Beware!