In Duplicate Bridge, does the person holding a particular set of cards open the bidding in all times that particular deal is played?
In pair competitions, yes absolutely ; everything about each hand is identical every time it is played. The dealer and vulnerability is determined by the board number usually - a sixteen board rotation covering all combinations of dealer and vulnerability. The holder has a number and the dealer and vulnerability marked on it.
See this picture from wikipedia:

The red sides indicate Vulnerable, and the dealer is marked as well.  Here NS are vulnerable and E deals. It will be played that way at each table.  Board 2 always has that configuration every time you play anywhere, as would board 18 and 34.
In Swiss teams it’s similar except you’d make sure the other team plays the opposite direction- so team A is NS at one table and EW at the other table, but the board is played the same. Thus one player is designated as Dealer (by the Board markings) at one table; and a player of the other team sitting in the same seat will be the designated Dealer when the Board is played at the other table.
Board-a-Match team games work the same as Swiss.