Isomorphic or Equal in Vect? In the category of vector spaces, how many elements are there in the isomorphism class of 1-dimensional vector spaces? Secondly, is the polynomial algebra generated by the symbol $x$ equal to the polynomial algebra generated by the symbol $y$, or merely isomorphic to it.

The first question can have no answer. Vector spaces over a given field (even $1$-dimensioanl vector spaces) are not a set, unless you consider a small category. It is for the same reason as the set of all sets does not exist.
For the second question, the answer is ‘yes: two polynomial algebras in one indeterminate over a given field $F$ are isomorphic’, since they are all realisations of the free algebra on one element. But they're not equal because the symbol $x$ (in general) does not belong to $F[y]$ and vice-versa.
