Let $X_1, X_2, \cdots, X_m$ be random samples from a normal distribution $N(\theta_1, \theta_2)$.
Then,
$$ L(\theta_1,\theta_2) = P(X_1=x_1;X_2=x_2;\cdots;X_n=x_m) = \prod_{i=1}^{m} \dfrac{1}{\sqrt{2\pi\theta_2}}{\text{exp}}{\Big[ -\dfrac{ (x_i-\theta_1)^2 }{2\theta_2} \Big]} \tag{1} $$
To relate to my doubt, I will shrink sample size to $m=2$. That is, we have $X_1,X_2$. Then,
$$ L(\theta_1,\theta_2) = f_{X}(x_1)f_{X}(x_2) = P(X_1=x_1;X_2=x_2) = \prod_{i=1}^{2} \dfrac{1}{\sqrt{2\pi\theta_2}}{\text{exp}}{\Big[ -\dfrac{ (x_i-\theta_1)^2 }{2\theta_2} \Big]} \tag{2} $$
But as per product distributions wiki, if $X_1$ and $X_2$ are two independent continuous random variables, both described by probability density function $f_{X}$, then the probability density function of $Z = X_1X_2$ would be
$$ f_Z(z) = \int\limits_{-\infty}^{\infty}f_{X}(x_1)f_{X}(z/x_1)\dfrac{1}{|x_1|}dx_1 \tag{3} $$
My questions:
1) Does eq.{2} also represent case $Z = X_1X_2$? If so, how do I reduce {3} to {2}?
2) If not, how are {2} and {3} are not related?
3) Is product distribution of $Z=X_1X_2$ same as joint probability distribution $f_{X}(x_1,x_2)$? What are the relation between the two.
Context:
A related question was posted here, from which this doubt arose. Now I am confused, why eq {3} did not interfere in my MLE case.
I took above MLE example from here, page 260.