Are the elements of a module also called vectors?

No. Normally we do not have special names like this ...

Elements of a group are called groupies

Elements of a ring are called ringlets

I wouldn't call this an answer, it is just my opinion.

The elements of a given vector space I would only call vectors, if the vector space is $k^n$.(Probably I am missing something...) Why? For example I would not call matrices, polynomials or functions whose codomain is a vector space vectors.

On the other hand given a generic vector space I would call the elements vectors. For instance I apply a norm to vector.

I am tempted to say I call vectors $v \in V$ a vector if $V= \bigoplus k$. Now there is alway an isomorphism so for me it is legit to call elements of a generic vector space vectors, but given a specific vector space it should be "canonically" $\bigoplus k$.

Now if I transfer this to modules over a ring $R$, I think it is fine to call elements of modules of the form $\bigoplus R$ vectors, but in general I would not call the elements of a module vector.