The absolute value of multi-index (confused)

For a multi-index $\alpha=(\alpha_1,\alpha_2,\dots,\alpha_n)$ the absolute value is $$\lvert \alpha \rvert=\alpha_1+\alpha_2+\cdots+\alpha_n$$

But, because it is the absolute value, shouldn't it be $$\lvert \alpha\rvert=\sqrt{\alpha_1^2+\alpha_2^2+\cdots\alpha_n^2} \qquad ?$$

Both of $\alpha_1+\alpha_2+\cdots+\alpha_n$ and $\sqrt{\alpha_1^2+\alpha_2^2+\cdots+\alpha_n^2}$ are formulas that may be useful in different cases.
Apparently you're reading a paper or text where the author decided that it will be useful for him to have a short notation for $\alpha_1+\alpha_2+\cdots+\alpha_n$. He is then perfectly entitled to define $|\alpha|$ to be that notation.
Do not think that simply because of the notation, this author's $|\alpha|$ has to have any particular of the properties of the length of geometric vectors you may be used to.
Note that $|\cdots|$ already has many different meanings, e.g. $|A|$ for the number of elements in the set $A$, or sometimes $|A|$ for the determinant of the matrix $A$ -- and you need to infer from the context which of the meanings is in play when reading any particular piece of mathematics.