I am trying to maximize the likelihood of a univariate gaussian for some $\mu$ and $\sigma$ given by below equation
$$ f(x) = \frac{1}{\sigma \sqrt{2 \pi}}e^{-\frac{1}{2}(\frac{x - \mu}{\sigma})^2} $$
My data $\mathbf{X} = [x_1, x_2] \in \mathbb{R}^2$ is producing high value for the joint density given by $$ \begin{aligned} f(x_1, x_2) &= f(x_1) . f(x_2|x_1)\\ \log f(x_1, x_2) &= \log f(x_1) + \log f(x_2|x_1) \end{aligned} $$
However, if I plot my 2D data with 256 data points (all of which outputs a high likelihood value), I get this plot which does not look like a Gaussian distribution, which should ideally look something like this. Ignore the color coding in both these images.
My question is:
What can be the possible reasons where the Gaussian likelihood for the data is high but it does not actually belong to a Gaussian distribution? Please note that this is a general question regarding the scenario where high likelihood does not actually imply the data belongs to the corresponding distribution. I am not looking for answers specific to my data $\mathbf{X}$ for which I haven't provided any details.
Any help is appreciated.
Update
I have added 3D html plots of $[x_0, x_1, f(x_0, x_1)]$ for an ideal case vs my case (You might need to download the html file and open in browser). The $f(x)$ is large for my plot and yet it does not form a normal distribution as expected. FYI $f(x)$ is in terms of logarithms.