What are examples of vectors that are not usually called vectors? In algebra, a vector is an element of a vector space. An example of such an element is a matrix.
In linear algebra, a vector is a shorthand name for a $1 \times m$ or a $ n \times 1 $ matrix. (Whereas a matrix itself is also a vector, by definition, but rarely referred to as such.)
In (analytic) geometry, a (euclidean) vector is a geometric object with a magnitude and direction. These can be represented by tuples.
Both matrices and tuples are clearly vector elements in some spaces, but are for some reason not called vectors by name, unlike euclidean vectors and row vectors.
Are there any other examples of vectors that are not called vectors, like matrices and tuples?

The quality of a "vector" you are describing is the ability to write it as a set of coordinates with respect to some basis of the space in which they reside.  In finite dimensions, provided that you maintain the ordering of the basis elements, this takes the form of a "tuple", or a row or column matrix. This "picture" of a vector space gets somewhat complicated once you move past finite dimensions, since our basis now contains infinitely many elements. 
Sometimes we can still write the vectors in our space as some kind of ordered list (like the space of all real sequences) even though the vector space is infinite dimensional. However we cannot write down a basis for these spaces, but we know that one must exist (assuming the axiom of choice).  For such vector spaces, the basis becomes fairly useless, and we have to adopt other tools to study them.
There are many examples of vector spaces which do not "look" like vectors in the traditional sense. The set of functions from a given set into the real or complex numbers will do, where we take the "pointwise operations" of addition and scalar multiplication.
