I say everything here from the perspective of someone who helped title Fist of the North Star VHS tapes back in the day so… mostly love for most of the anime community. 
This is where anime collectors and regular movie\TV collectors clash… organization. Your screenshot of “AMAZING”? That’s what my anime browsing Plex looks like so it’s doable - with some effort because anime fansubbers\rippers are freakin’ esoteric on purpose. Do you know how many codecs and font packs you had to install just to watch these things before VLC or even CCCP was a thing? And SSA subs? C’mon.
First let’s start with the built in standard options to comply with standard Plex build (and honestly most media builds like Jellyfin, XMBC, Emby… pretty much everything).
You’re going to have to split Anime TV series and Anime Movies up. There’s no “Anime Library” that mixes and matches both. No way around that unless you want to go DLNA and you don’t want that option if you want pretty (okay you can but that’s beyond the scope here). That mass of folders? Not going to work at all that way and no way to force it to work - it’s going to take some effort.
You can absolutely put Anime mixed in with your regular TV and Movies if you like… I put my anime movies in my regular movies library. I have a dedicated Anime TV library though - for various reasons.
I’m pretty well caught up on my series and I don’t really “collect” anime so… I don’t usually keep it once I’m done with it. I got outta that habit with anime a long time ago so that’s just me.
Movies are pretty easy actually so we’ll do series first. These are the naming rules you have to follow for TV shows: https://support.plex.tv/articles/naming-and-organizing-your-tv-show-files/
Same is true for anime.
Where ever your TV library is create a folder for your first Anime series. Lets use Durarara. Start by renaming that folder to “Durarara (2010)”. Why? Because that’s the title at TMDB (Durarara!! (TV Series 2010-2016) — The Movie Database (TMDB)) so that’ll match. That also is how it’s named at TVDB (デュラララ!! - TheTVDB.com) so double helpful. Adding the year can help with matching.
Note on TMDB vs TVDB - I like TMDB but pick what works for you. TVDB had alternate order for episodes that can be helpful in some situations and luckily you can mix and match per series.
Which one is used is set in your library or series in the options:
So here’s where we have a hiccup - your folder says “25” episodes but there’s only 24 episodes listed… something is off. So… file cleanup. Make sure your files match TMDB or TVDB listings for Season 01. Now you have to create a Season 01 folder and add your files to that folder and name them along the lines of “Durarara S01E04 Title of Episode”.
They won’t match otherwise. If they don’t match they won’t be pretty. You have some flexibility here between TMDB and TVDB episode ordering so you can pick TMDB or you can pick TVDB Aired, DVD or Absolute. If you use Absolute you still do S01EXX but EXX is just increasing.
Note - you can sometimes cheat with Season 01 file numbering. You can skip the season folder if it’s the only season and it’s numbered 01-57 or whatever. It’s not right… but it works most of the time… and if you want to absolute order it sometimes you can get away with it this way. It’s NOT RIGHT… but it sometimes works. Use at your own peril.
So… now it’s time for Durarara season 2! Create a Season 2 folder and now you’ll add your Season 2 episodes and rename them to match your source as well. Done!
You can mix and match episode order on shows - not seasons - so if you have multiple seasons see which order from TVDB or TMDB works best for ALL seasons.
Honestly… that’s it. Make your file names S00E00, match the ones you see at TMDB or TVDB, and use the expected TV folder structure and you’re done. As I said, I prefer TMDB to start and then use TVDB when I need an alternate order.
Here’s how mine end up:
Yeah… that file naming gets tedious. That’s the biggest headache.
You can do it manually (which is what I usually do because I don’t have tons to rename and I cheat sometimes) but otherwise folks recommend FileBot or TinyMediaManager to help you automate and do this for you. TMM is free but FileBot is more flexible for anime naming.
Anime Movies… same as regular movie library management really but again just match up to TMDB or TVDB for naming. You can do one folder with all movie files or you can do movie folder->movie file system. Up to you there. https://support.plex.tv/articles/naming-and-organizing-your-movie-media-files/
Extra subfolders seems to cause more issues these days so … avoid that and if you have organizational folders add those as a “top level” folder to the library. So for example I split up Movie titles by alphabet so 1-G folder is added to my library, not ..\Movies\1-G. Make sense?
OVA … this gets awkward. This depends a bit and you’ll just want to match what TMDB or TVDB say they are and go with that - sometimes it’s a movie and sometimes its a series (and TVDB can sometimes be both). But they work the same - just match.
This is where Black Lagoon was a problem, for example, because season 3 and some episodes were released as OVA and naming was all over the place… but flipping to TVDB DVD order for Black Lagoon worked out so I could keep them all together where as TMDB wanted to split them up between TV Series and Movie.
There is a bit of an alternative that’s outside the norm if you want… you can look into HAMA as an Anime specific scanning agent. That seems like a lot of extra effort but anime fans seem to love extra effort so maybe that’s okay (remember the codecs and also SSA SUBTITLES).
Really… FileBot is going to be your best solution here I think and then your Anime TV Library and Anime Movie Library will look great!
Oh… depending on your Library settings it’ll be better to rename before adding to your library. Adding and then renaming can cause extra effort so just rename first and then add to library.
Mixing and matching “lite” option. If you name a collection in a TV library the same as a collection in a Movie library they’ll kinda link. Here’s an example of my old Macross collections. They don’t work exactly the same (no combined watch order) but it’s something anyways.
Hope that helps you out … it’s mostly going to be about the file renaming effort. Once you get your batches done though it’s not that big a deal and if you have FileBot that can automate a lot of it for you.
I’m sure others will have some more advice … like I said… anime fans are esoteric. 