As it turns out, I actually researched this phenomenon the other day while doing some reading up on 旧字体【きゅうじたい】. As it turns out, what you're referring to are 書換字【かきかえじ】. Essentially, with the promulgation of the 当用漢字【とうようかんじ】 in 1946, the Japanese government decided to try to encourage some additional, more informal simplifications to bring vocabulary that depended on 表外字【ひょうがいじ】 into compliance with the new set without the need to resort to 交【ま】ぜ書【が】き (aka. substituting katakana for missing characters, such as 天プラ instead of 天麩羅).
As for why the need to create a second tier of simplifications instead of simply declaring these additional 新字体【しんじたい】 forms, I imagine a fair bit of that stems from maintaining overall linguistic integrity. The 新字体【しんじたい】 characters were based off of long-standing traditions and 略字【りゃくじ】 (abbreviated forms of characters used in handwriting) forms of their corresponding 旧字体【きゅうじたい】 characters. As such, there is a degree of continuity in terms of meaning.
In the case of 書換字【かきかえじ】, the substitutions are based primarily on substituting characters that are within the 当用【とうよう】・常用漢字【じょうようかんじ】 for ones outside of it. Oftentimes this involved preserving a phonetic element and just changing the accompanying semantic radical (e.g. changing from 骼 to 格). Other times it was something a bit more dramatic (such as changing 誨 to 戒). In some cases it involved simply removing parts altogether (e.g. 廻, 蛔, and 洄 all became 回).
With that foundation laid, let's touch back on the questions at the end of the original post:
Does this replacement of characters have a name? They are not shinjitai since they already existed and they are not in shinjitai tables.
As mentioned above, these are called 書換字【かきかえじ】.
Does this only happen with ateji or also with regular words?
The list I link at the end shows a great number of regular words. The primary reason for this was to bring words using characters that were left out of the 当用漢字【とうようかんじ】 list into compliance. As a matter of fact, there are a couple of rather common words that were affected by it (e.g. 注文【ちゅうもん】 was originally 註文【ちゅうもん】)
Does it only happen when the replacement characters have a same reading as the replaced characters?
Yes, all characters have the same reading as the ones they replaced, at least for the purposes of the word in which they are serving as a replacement. Another feature that distinguishes 書換字【かきかえじ】 from 新字体【しんじたい】 that I've observed is that from what I can tell the 書換字【かきかえじ】 were simplified as words, whereas 新字体【しんじたい】 were simplified as characters.
If not, does this add to the confusion of which readings apply to which characters?
As mentioned above, they generally aren't adding any new readings to the "simplified" characters. So no, it doesn't add to the complexity in that regard.
If you want an overview of just what was changed and how far-ranging these changes were, this is the most comprehensive list I've been able to find. For simplicity, the columns from left to right are:
- The modern reading of the character replaced
- The classical reading of the character replaced
- The characters involved (new on the left, old on the right)
- The authority promulgating the change (either the government 国 or newspapers 新)
- Affected words
- Additional notes
This might have been a bit more information than you were looking for, but I hope it helps!