Hm, if you registered a taxonomy for the “page” object type correctly and then assigned a term of that taxonomy to a page… I believe you can then access the taxonomy and term slugs in the following way:
get_query_var( 'taxonomy' )
get_query_var( 'term' )
If you print_r($wp_query) you will see all the parameters that are there when generating a current page that’s displayed. With code above you’re accessing those parameters from $wp_query.
Then to get the term object with full info you can use get_term_by function, like so
$term = get_term_by( 'slug', get_query_var('term'), get_query_var('taxonomy') );
echo $term->name;
This will print the “nice” name of the term.
I believe if you use get_query_var('term') or $term->slug (after getting the term object) you can use that slug in all of other queries.
Hope that helps. I never used taxonomy for pages.
Let me know how you get on.