All positive and negative numbers including zero are called integers. So in the form $a=bq$, since $0 = 0ㆍq$ is true for any integer $q$, $0$ can have $0$ as a divisor of itself as well as a multiple of itself by the definition expressed by $a=bq$.
But why it's said "We cannot divide by $0$"? It's understood as "$0$ cannot be a divisor" to me.
"Definition: An integer a is called a multiple of an integer $b$ if $a=bq$ for some integer $q$. In this case we also say that b is a divisor of $a$, and we use the notation $b | a$ . . . On the other hand, for any integer $a$, we have $0 = aㆍ0$ and thus $0$ is a multiple of any integer."
Source: Abstract Algebra: Third Edition, John A. Beachy, William D. Blair, p.4.
"Rule Division by $0$ is undefined. Any expression with a divisor of $0$ is undefined. We cannot divide by $0$"
Source: Prealgebra: A Text/Workbook, Charles McKeague, p.61.
"Observe that division by the integer $0$ is not defined, since for $n≠0$ there is no integer $x$ such that $0ㆍx=n$ and since for $n =0$ every integer $x$ satisfies $0ㆍx=0$"
Source: Introduction to Mathematical Proofs, Second Edition, Charles Roberts, p.99.
[Now I understand my question more after reading number theory chapter of a book]
$0=d\cdot 0 $
Thus, 0 is a multiple of every integer except 0.
Best Answer
Every integer divides zero, including zero itself; however, the only integer that zero divides is itself. That is, $b \mid 0$ for all integers $b$; but if $a$ is an integer and $0 \mid a$, then $a = 0$.
When it is said that "you can't divide by zero", what is meant is that, given an integer $a$, there is not a unique quotient upon division by zero.
Specifically, given integers $a$ and $b$, with $b \ne 0$, if $b \mid a$ then there is a unique integer $q$ such that $a = qb$, namely $q=\frac{a}{b}$. However, if we allow the case when $b=0$, then we lose the uniqueness. Indeed, as already mentioned, $0 = q \cdot 0$ for all integers $q$, so it makes no sense to assign a value to the expression $\frac{0}{0}$. And if $a \ne 0$ then there is no integer $q$ such that $a = q \cdot 0$, so it also doesn't make any sense to assign a value to the expression $\frac{a}{0}$.