Note that Stack Exchange is using a 3rd-party library highlight.js for syntax highlighting, and looks like it currently doesn't support LilyPond syntax.
Stack Exchange does not have its own syntax highlighting engine. It uses highlight.js, and may not necessarily be using the latest release of that library. Therefore, any bugs and feature requests regarding syntax highlighting cannot be handled by Stack Exchange.
So, before requesting to enable the existing syntax highlighter that will possibly not work effectively anyway, you might try to request highlight.js to add support for LilyPond first.
Requesting that a new language which is not currently supported by highlight.js be added
Follow the instructions in Reporting a bug or requesting a feature for a language supported on Stack Exchange above to request that highlight.js support your new language. Once the language is supported by highlight.js, follow Requesting that Stack Exchange support a new language which is already supported by highlight.js to make it available here on Stack Exchange.
Which is
Reporting a bug or requesting a feature for a language supported on Stack Exchange
Go to the list of existing issues in highlight.js GitHub repository and check to see if your request is already filed there, and if not, post it there.
If your bug fix or feature request has already been fixed or implemented by highlight.js but isn't working here, please wait, as new versions of highlight.js are deployed on the sites on a scheduled basis, and the latest version hasn't been deployed yet. (If it hasn't been deployed after a long time, you can post a feature request here on Meta asking that Stack Exchange update to the latest version.)
then,
Requesting that Stack Exchange support a new language which is already supported by highlight.js
If a language is already on the list of languages supported by highlight.js, but is not supported on Stack Exchange (see Language codes currently available on Stack Exchange below), you can raise a feature request here on Meta to ask for it to be deployed on the network.
While it's fine to request for the SE team to add additional supported languages, these requests are extremely likely to be considered low-priority and as such deferred. The current response for such feature requests is that the team finds that the added trade-offs from supporting more languages outweigh their potential benefit, though they're looking for future ways to lighten it.
Otherwise, the community may be better to suggest a ready-to-use LilyPond syntax highlighter library that can be embedded on the website since it is very unlikely for the Stack Exchange team to create a new syntax highlighter specifically for LilyPond from scratch.