I host my own personal manga server that I use tachiyomi on my tablet to read from it with. The server itself does tracking, but I also plugged tachiyomi into MAL, so I have 2 lists acting as a backup of each other.
For anime I just let my media server track it, keeps track of every season/episode play status. When a new season of something starts it pops up ready to be watched.
I use tachiyomi but what is this media server thing you are referencing? First time hearing about this in this context, I would have thought it was like having a cloud in your home you can pull files from over LAN/internet but I dont understand how it helps with tracking. Just from what it sounds like, you download the files, create the entries (else how would you track?) and then some software tracks what you have already read/seen? If thats what it is do you delete the media after you watch/read or would that mess with the entries?
First time hearing about this, I knew tachiyomi could be used as a reader to pull local files instead of using the extensions but this kind of use has never occurred to me.
For manga the server is komga, and i use tachiyomi to connect to it to read. Komga does its own tracking, and I also added MAL as a tracking in tachiyomi itself.
For anime (and everything I watch, also movies/tv etc) I use jellyfin. It keeps track of everything you watched. So if I finish season 1 of something and a year later season 2 releases, once I have the first episode of season 2 it'll pop up on my list of things to watch. There's been series I forgot about until this happened.
what is this media server thing you are referencing?
take a look at plex or jellyfin for example. these are media servers that can be setup to automatically organize your media library and keep track of what you have downloaded, what series you are watching, what episodes you have watched, etc. also like you mentioned they also provide some sort of streaming service where you can watch your media over the network without having to download it on every single device.
I'd also like to mention that usually downloading is done using an external software like sonarr or radarr that fall in a gray zone in terms of legality so be careful if you ever plan to go that route.
for manga specifically, Komga can serve and track your progress. it integrates with tachiyomi and also runs a webapp that you can fallback to. i tried Jellyfin for manga and the UI just was not fun to use tbh.
neither of these track your progress publicly in the way that MAL does though, AFAIK.