Multiplication of two impulse function $\delta(t)\cdot \delta(t)=?$
The property

$$f(t)\delta(t-t_0)=f(t_0)\delta(t-t_0)\tag{1}$$

is only valid for a function $f(t)$ that is continuous at $t=t_0$. Since the Dirac delta impulse $\delta(t)$ is not a function (it is a distribution) and since $\delta(t)$ is not continuous, property $(1)$ does not hold (and does not make sense) for $f(t)=\delta(t)$.

The product $\delta(t)\cdot \delta(t)$ is undefined. Also note that the quantity $\delta(0)$ is meaningless. Since $\delta(t)$ is not a function you cannot determine its value for any value of its argument.