I think the Forum HTML cleaning is just more aggressive, and it's probably a combination of security plus Moodle HQ philosophy. For example, the following setting description suggests that indeed some content is treated differently than others in terms of what is stripped. Philosophically, allowing certain kinds of HTML behavior in Forum posts could lead to user confusion, or otherwise a bad experience. Of course other forum modules out on the internet have this kind of feature, but since it's not a thing in Moodle, people might not expect it. I'm not defending that philosophy, just speculating.
"Experimental settings
Content cleaning everywhere forceclean
Default: No
Content
added to the site is normally cleaned before being displayed, to remove
anything which might be a security threat. However, content is not
cleaned in certain places such as activity descriptions, page resources
or HTML blocks to allow scripts, media, inline frames etc. to be added.
If this setting is enabled, ALL content will be cleaned. This may result
in existing content no longer displaying correctly."
If you are feeling adventurous, "Enable trusted content - enabletrusttext" in Site security settings may allow more HTML tags through in Forum posts. Definitely read the description. Further than that, you could probably hack it in the Moodle source code, but I'll leave that to someone else!