The number $\tilde{p}$ is said to approximate $p$ to $t$ significant figures if $t$ is the largest non-negative integer for which $$\left|\frac{p-\tilde{p}}{p}\right| \le 5 \times 10^{-t}$$ Source
Now, suppose the actual value $p$ is $1.0$ and that the approximation $\tilde{p}$ is $1.9$. Then $$\left|\frac{1.0-1.9}{1.0}\right| \le 5 \times 10^{-t}$$ $$0.9 \le 5 \times 10^{-t}$$ $$10^t \le 5.\overline{55}$$ $$t \le log_{10}(5.\overline{55})$$ $$t \le 0.74473$$
From the conditions ($t \in \{0,1,2,3,...\}$ and $t \le 0.74473$), $t = 0$.
But doesn't $1.9$ approximate $1.0$ to one significant figure? (they both have a leading one)
Whether this definition will follow your intuition depends on where you draw $p$ from. For example, with $p = 1000$, $\tilde{p} = 990$, you get $2$ significant figures. And yet their graphical representation does not have $2$ digits in common.
Maybe it was meant for $0 < p < 1$, and not any non-zero real number?