Yes, that's possible. As mentioned above MB does not explicitly support genres, but folksonomy tags can be used for it. Picard itself has basic support for this using MB folksonomy tags, alternatively you can use the Last.fm plugin from https://github.com/fdemmer/Picard-Last.fm.ng-Plugin.
There is also a plugin in the works to pull genre information from Wikidata, not sure how usable it already is.
Regarding tagging only specific tags you can configure Picard to preserve specific tags in the options. There are already a couple of topics here discussing this.
I think your comment is misleading, as e.g. Picard can use the folksonomy tags for genre tagging and genre was one of the leading considerations when folksonomy tags where introduced. There are many use cases for folksonomy tags, but genre is maybe the most important one.