$A$ is unital. Something a little more general is true: if a C*-algebra $A$ has an element $b$ that is a right divisor of all its elements, then $A$ is unital. Proof: We have $(b^*b)^{1/4}=xb$ for some $x\in A$. So
$
(b^*b)^{1/2}=b^*x^*xb\leq \|x\|^2(b^*b).
$
The inequality $(b^*b)^{1/2}\leq \|x\|^2(b^*b)$ holds in the C*-algebra generated by $b^*b$, and this implies that 0 can only be an isolated point of the spectrum of $b^*b$. So the C*-algebra $C^*(b^*b)$ has a unit $p$. Since $b(b^*b)^{1/n}\to b$ and $p$ is a unit for $(b^*b)^{1/n}$, we also have $bp=b$. Any other element of $A$ is a left multiple of $b$, so $p$ is a (right) unit for $A$.