Whottt, I have talked about this subject numerous times in here. I don't know Tim's internal mindset of why he wants to be listed as a PF, but since he left Wake Forest, he was brought in to play the PF (4) beside David and David was always listed as a Center (5). Tim is not a leaper, so he's never jumped that well at the opening tip against some of the 7 footers, a traditional Center function.
As long as Shaq is playing, he will be considered the top Center because of his sheer force in the paint. Tim has a fair chance at being considered the top player at his position if he is officially listed as PF. So maybe, that's the reason Tim likes to be listed as such. Managing perception is as important a factor as anything in many areas of life and not just in sports.
The reality is that the 4 and 5 positions in many offenses are basically the same function. It provides for more flexibility on ball reversals and moving the defense around when you can present posts on either side or high and low. Henry Iba's motion offense passed down through the generations to the Knights, Suttons, etc. and motion-based offenses like the Flex, Princeton, or even the Triangle can have both the 4 or 5 as post players.
When a post player can back screen out on the floor (15-18 feet) for a wing player and have the range to shoot from there like Duncan can, it changes the strategy as to who defends them, thus Duncan tends to draw the smaller of the two bigs on him.
Tim has such a strong base and big backside that he can play with his back to the basket and play a traditional pivot's role. Pop uses Tim as both a 4 and 5, and it often depends on who's in the game, offensively and defensively.
The Jump Center, for which we get the shortened name "Center" (back when a jump ball was held after every made basket) has morphed over time into the tallest or one of the tallest players with an inside presence. It's not as clear-cut as it used to be.
In fact, a case can be made for the Spurs using Tim as a Center/Pivot the way the Lakers used to use Mikan as a Center in the pivot, as the pivotal hub of a weaving and pass-to-the open man offense. Even though Pop may call it 4-down on the block and there's a little less weave and more cut through and move for spacing, he is still the "Center" of the offense. But that's playing with the name Center and not really using it in it's original context.
It's all a management of perception and not necessarily reality. Tim is a post/pivot player but officially he wants to be called a Power Forward. Whatever...