Hi,
Yes, it is possible to do it indirectly via term metadata, then enabling search in the metadata fields.
The easiest way to add a term metadata field non-programmatically is by using ACF plugin. If you are already using it, it has a feature to add term meta fields via their UI. It is also possible to do it programmatically, for that please check this tutorial.
As for the second questions, the reason is that the phrase does not match the product category title nor the description only the product fields.
However if you want to, you can display the product categories (and many other things) within each results via the advanced content field.