First, MC doesn't handle Season names at all. Just Series and Episode names, and Season and Episode numbers.
Next, it is always best to follow whatever is done on TheTVDB, as that is where the data comes from for TV series. Mostly anyway.
So if you look at the American Horror Story record on TheTVDB,
https://www.thetvdb.com/series/american-horror-story, and specifically say at the Season 1 details,
https://www.thetvdb.com/series/american-horror-story/seasons/1, you will see there are various poster images available for the Season, and many have the Season name on them.
So, in Standard View, select the American Horror Story series thumbnail, right-click > Cover Art > Get from internet, then you will have the opportunity to select from the available Season Cover Art images for each Season. Pick one that includes the Season Name, such as "American Horror Story Murder House".
That is what I would do anyway. That way, in Theatre View, all of "American Horror Story" stays together as the one Series, under one thumbnail image, and under that each Season is displayed with its Season name in the Season thumbnail image.
Note: If you really wanted to see the Season name as text under the thumbnail, you would have to create a custom field, perhaps named [Season Name], and then edit the thumbnail text to display that field. But you would have to populate that custom field manually for all Episodes in a Season, and then use the text from an Episode for display. You can't save a [Season Name] just to a Season, because MC doesn't actually store any information by Season or Series. Everything is stored by file, which means Episode.
PS: Do you really use MC Standard View for playing TV Shows? Theatre View is so much better for displaying the Series/Season/Episode relationship. Standard View just shows a thumbnail for each Series, and the list of Episodes sorted, usually, by Season and then Episode number. Nowhere near as nice, but good for maintenance.