Japanese here. There are specific editors often categorized as "アスキーアートエディタ" (Ascii Art Editor). They often come with libraries of existing AA to modify, and can display image with low opacity to "trace" image.

Large part of jp ascii art made/appeared in 90s to early 00s at underground forums like 2ch. Those editor were tailored for their speifics such as font family (MS-P Gothic) and size. The display/font system is different nowadays, but unlike "AA artist" in those times who is 100% comitted to make vertical line aligning pixel perfect most of people find those AA displays ok.

First off, MS Gothic is a monospace font. (Meaning all characters have the same width and move the cursor by the same horizontal distance. Ok, that's a slight oversimplification. Especially when you're dealing with asian characters, there's a possibility of "double-width" characters which are twice the width of "single-width" characters and move the cursor twice as far.) In sites like Lemmy, there are usually ways to tell Lemmy to switch into "monospace" like you did with the first cat art you posted in the body of your post. And that ensures consistency in the output. With non-monospace fonts, it's more of a crapshoot. Arial's "m" might be a different width than Comic Sans', for instance. Typically, sites like Lemmy (or 4chan or Reddit or Facebook or whathaveyou) won't have ways to specify a particular font (different Lemmy clients are also free to use different fonts), so if you composed ASCII art with a non-monospace font and pasted it into a Lemmy post/comment, even if it looks right to you in Lemmy, it may not look right to other viewers of your post. And that's why monospace is popular for these things.

How to make these? I honestly don't know if there are specialized tools for that. Probably just a standard text editor. The examples you posted have some asian characters in them, so a way to input such characters. I'm on a Linux machine and have fcitx set up for Japanese text input. If you're on another OS, I'd expect the way to set up input for asian characters would be different. Alternatively, there are probably unicode character explorers/apps that can be used and don't require quite the learning curve.

As for how to manage these, no idea. I can think roughly about how I might go about writing a program that migth manage these, but I'm not sure if any exist out there currently.

I don't think, there's a special trick to making them. You can look at existing kaomoji lists and pick out individual symbols to create the shape that you want.

Or you can combine kamojis. For example, maybe you want a cat handing over a flower, but you want it to look sad, like an apology.

Then you find a sad cat kaomoji:

/ᐠ • ˕ •マ

And combine it with the kaomoji you posted:

⠀/\__/\
 (• ˕ •)    
/ >🌷< \

Well, could be better, but just as an example. Combining different faces and arm shapes and such is relatively easy.

As for managing them, I usually see tags assigned to them. On the webpage that you posted, it's the little text boxes below the kaomoji.
But in its simplest form, you could have a text document and just write a few words above each kaomoji, like e.g. "sad cat flower". Then if you search the text document with Ctrl+F for "cat", this will be one of the results.

https://japaneseemoticons.me/all-japanese-emoticons/

Basically off topic but... At one point I tried generating some mouse ASCII art with Ch*tG*T and

  ,     ,
  (\,;,/) 
   (o o)
==\_~_/_==
 /   \    
 \   /    
  `"`"`   

and I really liked it

  ,     ,
  (\,-,/)
   (o o)
 ===\_/===
   / o
  / mm\
(_\   /
   `"`"

And now I just want to know where it stole the original from, so I could give credit when I use it...