Making the changes you want will likely require significant customization - beyond the level our team can help with via the forum, unfortunately. I can try to give you some initial pointers as to where to look to find code related to this, but you will likely need to figure out what to modify where yourself.
In Elasticsearch 5.6, I believe that the facets are powered by what is known as aggregations:
I am not 100% sure, but I think what you will want to look at first will be term aggregations:
In AtoM, much of Elasticsearch is configured via the arElasticsearchPlugin - this is where you will find most ES-related code and index mappings:
For example, the mapping of what AtoM fields are indexed (and how) is in this configuration file:
While the ES analyzers and tokenizers used are defined in this configuration file:
Meanwhile, there are functions defined in the plugin related to handling aggregations - you can search for "aggregation" or "aggs" in the code repository to find some relevant spots. For example, here are two functions related to aggregations:
The general structure of the factets as a sidebar is defined in a partial that is reused in many places in AtoM - see:
Now, on the actual Authority record browse page, here is the browse template defining the facets (aggregations) shown:
Then there is the styling of these, in the relevant theme plugin - here is the Actor (i.e. authority record) browse page template in the newer Bootstrap 5 Dominion theme:
Unfortunately, beyond finding these starting points, that is all the help I can personally provide at present. I will see if the developers have any additional suggestions for starting points, but my suggestion would be to first understand how aggregations work in ES 5.6, then figure out how we've used them in AtoM - that should help you figure out what is possible, and eventually what to change.
Good luck!