Figure is shown incorrectly

### Describe the issue linked to the documentation

Second part of the [figure](https://scikit-learn.org/stable/modules/gaussian_process.html#dot-product-kernel) doesn't show sampled functions

### Suggest a potential alternative/fix

_No response_

COMMENT:
My understanding of Gaussian Process is very limited, but I'll write down what I found in case it could help someone solve this.

It seems that the convergence problem was present from the beginning and has appeared in all the releases ([v0.18](https://scikit-learn.org/0.18/auto_examples/gaussian_process/plot_gpr_prior_posterior.html), [v0.19](https://scikit-learn.org/0.19/auto_examples/gaussian_process/plot_gpr_prior_posterior.html#sphx-glr-auto-examples-gaussian-process-plot-gpr-prior-posterior-py), and [v0.21](https://scikit-learn.org/0.21/auto_examples/gaussian_process/plot_gpr_prior_posterior.html) don't show the warning in the rendered notebook but I ran the examples with versions `v0.19` and `v0.21` and they did appeared. For `v0.18` I couldn't replicate the environment quickly.). The warning appears only for the dot-product kernel.

In order to avoid getting the warning, I added two parameters to the `L-BFGS-B` optimizer, either one works:
- Increase the maximum number of line search steps (per iteration) to 40 (default is 20).
- Remove the bounds.

Thus, we could edit the code for the GP with dot-product kernel like this:
```python
import scipy.optimize
from sklearn.utils.optimize import _check_optimize_result
def my_optimizer(obj_func, initial_theta, bounds):
    opt_res = scipy.optimize.minimize(
        obj_func,
        initial_theta,
        method="L-BFGS-B",
        jac=True,
        bounds=bounds,
        options={'maxls': 40},
        # bounds=None,  # This is the other option without increasing the number of line search steps.
    )
    _check_optimize_result("lbfgs", opt_res)
    return opt_res.x, opt_res.fun

from sklearn.gaussian_process import GaussianProcessRegressor
from sklearn.gaussian_process.kernels import ConstantKernel, DotProduct

kernel = ConstantKernel(0.1, (0.01, 10.0)) * (
    DotProduct(sigma_0=1.0, sigma_0_bounds=(0.1, 10.0)) ** 2
)
gpr = GaussianProcessRegressor(kernel=kernel, random_state=0, optimizer=my_optimizer)

...
```

Nevertheless, the GP fit does not appear to be good. It actually is fairly similar to the GP that raise a warning.

With warning the GP has:
```
Kernel parameters before fit:
0.316**2 * DotProduct(sigma_0=1) ** 2)
Kernel parameters after fit: 
0.674**2 * DotProduct(sigma_0=2.13) ** 2 
Log-likelihood: -7957695978.947
```
![imagen](https://user-images.githubusercontent.com/2133361/226037114-a1a816cf-c818-46eb-907e-d74f4b6dc5e5.png)

Without warning the GP has:
```
Kernel parameters before fit:
0.316**2 * DotProduct(sigma_0=1) ** 2)
Kernel parameters after fit: 
0.674**2 * DotProduct(sigma_0=2.13) ** 2 
Log-likelihood: -7957695978.947
```
![imagen](https://user-images.githubusercontent.com/2133361/226037357-e6e3223d-2fd0-4ec8-9e65-b3bc7d192913.png)


On the other hand, the mean function from the GP posterior rendered on the notebook is different from what I got when I run it.
I run the [code to replicate the graphs](https://github.com/scikit-learn/scikit-learn/blob/main/examples/gaussian_process/plot_gpr_prior_posterior.py) and these were the results:
![imagen](https://user-images.githubusercontent.com/2133361/226001400-d71418bc-44fe-45cd-a318-47971e9e8c51.png)

While the [rendered notebook](https://scikit-learn.org/dev/auto_examples/gaussian_process/plot_gpr_prior_posterior.html) shows:
![imagen](https://user-images.githubusercontent.com/2133361/226001985-ec954602-407e-4c58-a13d-f6a8a50c65b8.png)

Is this behavior expected?

Labels: Documentation, Needs Investigation