This is a Post-War mfg Walther PP I assume.
No 'P' after the ser# on the frame and a Ulm, W Germany address on the slide.
These use the so called 'round' style firing pin.
They are refering to the section inside on the pin that is engaged by the rotating safety, not necessarily the end that you can see just by looking at it.
But..that Round end of the firing pin you can see DOES indicate that the firing pin is of the round profile inside where it engages the Safety.
The earlier style (War time production and Pre-war) used the 'Flat style' or sometimes called the Square style firing pin.
Again that name indicates the shape of the engagement lug on the pin itself inside where it locks into the Safety. It is also the shape of the end of the firing pin that you can see. So again it tells you quickly what type you have.
The 'Round Style' was a Post war re-design feature on the Walthers.
It was used from the beginning of post-war production straight on through to the end of (W)German & French (Manurhin) produced pistols.
Each type uses their own thumb Safety as the machine cuts inside to engage the different styles are needed.
(There are different Pre-War editions too, such as the early 90* safety and I believe the very first pistols using a 2 piece firing pin had their own thumb Safety machining. )
So,,If the orig Post war Safety in the pistol is OK, and you just need to replace the firing pin,,then you need that PostWar 'Round Style' centerfire pin for Model PP.
IF the firing pin is OK, and the original mfg thumb Safety is bad,,then you need an orig Post War Thumb Safety as a replacement to work with that orig Firing Pin.
*** Here's where the problem arises when 'new' parts are used...
When the Walther model PPK and PPK/S were licensed to be 'Made in the USA,,,some minor changes were made in the design.
One of the changes was to the firing pin..
If you order a new firing pin these days for a Walther Model PP or PPK or PPK/S,,you most likely get one of the new designs.
That makes 3 different ones counting the 2 above.
This third one works just fine in the Walther/German Post war mfg pistols,
BUT, it requires an American redesigned thumb Safety to work along with it .
The main different in this newest design is the full circular guide lug ahead of the engagement portion of the pin.
The two previous designs had a small square lug to serve as a guide.
So the common fix is to change out both the firing pin AND the Thumb Safety if there is a problem with either of the parts.
Here's a link to a gun parts seller
I only link it because of the pics.
It shows all 3 styles
Walther PP, PPK, PPK/S .22, .32, .380 pistol – Jack First Inc.
Pg 1 has:
The old style square (PreWar and War time)(square engagement lug & square end. Square guide lug.)
Post War German Round type (has round engagement lug and round tip, Square guide lug).
Click on the 'out of stock' pics for a clearer and larger image.
pg2 has:
Newer US redesigned (Round engagement and round end. Full circular guide lug up front)
Fits PPK, PPK/s & PP