Most probably your notes don't say that a vector is in itself coordinate invariant, but that a particular expression involving vectors is coordinate invariant. That's a property of the calculation encoded by that expression, rather than one of the vector that results.
An expression (or property) is coordinate invariant if all of the operations in the expression are ones that don't need to change if you decide to use a different coordinate system to express your vectors.
For example, $v+w$ is a coordinate-invariant expression, because vector addition is the same no matter which coordinate system we work in. Conversely, $(v_1+w_1, v_2+w_2, v_3+w_3)$ is not coordinate-invariant, because picking out particular coordinates of the vectors depends on which coordinate system you're using.
The term is somewhat fuzzy in that we need to agree first which coordinate systems we want to preserve our right to choose between. For example, the dot product is a coordinate-invariant operation when we consider only rectangular coordinate systems with a common scale on the axes, but not if we consider arbitrarily skewed and/or scaled coordinate systems.