Given two sequences of the same size, how many longest common subsequences can they have? For simplicity assume that both have the same size N.

the lengh of this subsequence can be at most N, so maybe it's

max(C(N,1), C(N,2), ... , C(N,N))?
I understand the question as follows:

Given $n$. How many different longest common consecutive subsequences might two strings of length $n$ have at most?

If this is your question, than consider for example the string $a_1a_2...a_n$ with $a_i\ne a_j$ for $i\ne j$ and the reverse string $a_na_{n-1}...a_1$. The longest common subsequences of those strings are all subsequences of length $1$ $(a_1,a_2,...,a_n)$. Therefore, in your unrestricted question, the maximal number of different longest common subsequences of two strings of length $n$ is equal to the the maximal number of different subsequences with same length, which is $n$ for subsequences of length $1$.

If you restrict the question to strings over a given alphabet of fixed size, lets say $k$, the answer is much more difficult.