Is the standard definition of vector wrong?
The general definition of vectors do not require that a vector has a defined magnitude. It just so happens that in many of the common vector spaces that are useful for physics and sciences that there is a natural choice for a norm, making these spaces normed spaces (which are a special kind of vector space).
In general a vector is merely an element of a vector space.
A vector space is a set $\mathbb{X}$ such that the following properties hold:
- For every $x,y\in\mathbb{X}$, you have also that $x+y\in\mathbb{X}$ (i.e. it is closed under addition)
- For every $x\in\mathbb{X}$ you have $\alpha x\in \mathbb{X}$ where $\alpha$ is an element of an underlying field (commonly $\mathbb{R}$ or $\mathbb{C}$) (i.e. it is closed under scalar multiplication)
- There is a "zero element", $0\in\mathbb{X}$, such that for each $x\in\mathbb{X}$ you have $0+x=x=x+0$.
Also required are things about the commutativity and associativity of addition, and distribution of scalars over vectors as well as distribution of vectors over scalars.
That is not a very general definition of a vector, no. But it is hardly "wrong" for that reason.
Direction and magnitude are intimately tied to spatial intuition.
The magnitude referred to is usually a "length" measured by a real number, but there are vector spaces where that idea makes no sense.
The same goes for "direction" too. You can, at best, only get a very primitive notion of direction, as discussed here
The best definition of a vector is probably "element of a vector space," with all the axiomatic behaviors implied. That does not mean it must be particularly useful for developing intuition about what a vector "is."
The direction-magnitude is a particularly apt way of describing vectors in 2 and 3 dimensional geometry, so it has its uses. After convincing yourself about higher dimensional real spaces, and the spaces over other fields, you will probably find yourself abandoning direction and magnitude in favor of the most general definition.