Is it a players whole turn if they are chosen for a Showdown?
I've never played this but I've now read the rules pdf from here
Firstly I couldn't find anything in the rules that indicated when the next player would start there turn!  I can only assume this the same as [basic Uno rules] in that play passes clockwise after a player takes a turn.2
The rules for showdowns say

The player who played the card takes the Showdown unit and places it
between themselves and the next player in the current order of play
(unless they played a Wild Showdown Card – more on that later).

and later

When you play this card, you get to choose which player to have a
showdown with

There is nothing in these rules which indicates that taking part in a showdown is a players whole turn.  It seems to me that a showdown just happens with the. next player in turn order and then that player takes there turn after.
If it was that players whole I would expect the rules for wild showdown to suggest that play continues from the player chosen for the wild showdown, which. they do not.
As a general rule for games, if something is not stated clearly in the rules they it almost certainly is not a rule.