Euclidean vectors (in a $2$-dimensional space) are defined as ordered pairs of real numbers, which in a Cartesian coordinate system can be graphically represented by a directed line segment going from $(0,0)$ to a point representing said pair.
Considering that the tuples themselves ARE vectors, it would seem that if we change the position of axes of the Cartesian coordinate system (so that the angles between $x$ and $x'$ and $y$ and $y'$ axes are $45$ degrees, for example), the vector, being a tuple, will now be represented by a segment going from $(0,0)$ to wherever the tuple its end was originally assigned to is now placed, leaving its position relative to the axes unchanged.
This is not what is happening. Instead, the line segment doesn't change its position and angle toward the original $x$ and $y$ axes, so when the coordinate axes move, it has a different position relative to new axes than to old ones. What's more important, the pair the line segments end on is now different - if we had a vector which originally was $(1,1)$, and move the coordinate axes $45$ degrees counterclockwise, that point is now $(1,0)$.
Once again, these vectors are defined by tuples. A tuple is a vector. How can we call the original $(1,1)$ and the new $(1,0)$ the same vector? Why isn't the $(1,1)$ in new coordinates considered the same vector as $(1,1)$ in original coordinates?