I did this. In one table visualization I put $Table, $Rows, and $Fields. In another table visualization I put $Field and $FieldNo.
I found that if I select a specific value of, say, $Table, the filter does not appear in the Selections bar at the top (it continues to show "No selections applied"). Nor can the filtering be undone, except by refreshing the web page. In Qlik Sense Desktop, not even refreshing the page undoes the selection, nor does closing and reopening the app, nor does closing and restarting Qlik Sense Desktop. The filtering becomes a permanent feature of the app!!
Also, if I make a tentative selection of a table name value, unselected and excluded values in associated fields in the $Field-$FieldNo table do not display. If I approve the selection, all the unselected values in the $Table-$Rows-$Fields table disappear as well.
In addition to the fields extracted from the data source, system fields are also produced by Qlik Sense. These all begin with "$" and can be displayed like ordinary fields in a visualization, such as a filter pane or a table. System fields are typically created during script execution, and are primarily used as an aid in app design.
System fields are not included in field lists in the assets panel or the Expression editor. If you want to use a system field, you need to reference it by typing it manually.
Available system fields
The following system fields are available:
Displays all internal tables loaded by the script.
Displays the fields that are read from the tables.
Represent the number of fields in different tables.
Displays the position of the fields in the tables.
Displays the number of rows in the tables.
Displays names of info tables, if they have been included in the app.