As far as I understand, fighter pilots or more general military aircrews will use a dedicated phraseology to communicate with each other, which is different to communicating to ATC. Is this phraseology standardized across different air-forces or even within NATO? Are there any public documents on this phraseology?
There are NATO standard terms. For example the MiG-29 is known as a "Fulcrum." That's not the Russian name for it but the "NATO reporting name" which you can look up on Wikipedia.
There are also some standard (but not codified) terms such as "angels" meaning "thousands of feet." And there are some standard terms such as "bent" meaning "inoperative" but I don't know how much those are used any more.
Within a certain operating area or task force, there may be specific code words used that may change from day to day. These would not be known to anyone outside that area or task force.
When I read this question, I think of the chit chat which goes on during sorties. It IS non-standard, and varies by location and service. On youtube there is a recording of a F-4D pilot who went into the drink at night in the North Sea.
Listen to that (long) recording. The dialog is quite similar to the dialog used, for example, by pilots in Vietnam. (A buddy of mine was base commander of the affected pilots.)
You may find more current examples from more current conflicts where there are videos with radio audio. Much of the terminology is not standardized, but there are certainly components which are.
An example of terminology which is rather universal but perhaps not standardized, might be the term "feet wet" which means the flight has crossed the shoreline and is over water.
They use brevity codes. They are used to quickly convey information on the radio. Like all things military, they are standardized but they do have some variation between services. You can find a list of US/NATO multi-service codes on Wikipedia.