The newly-formed West German bundeswehr briefly used this WW2-derived splinter camouflage smock shortly after West German reearmament in 1955 under Konrad Adenauer in response to escalating Cold War tensions. This camouflage pattern was only produced between February 1956 – February 1957 and was withdrawn from service in 1960, likely due to cost and its excessively Wehrmacht-like appearance (source: https://www.wehrmacht-awards.com/forums/forum/ken-jasper-international-militaria-forums/bundesrepublik-deutschland-1949-present/966033-early-bundeswehr-uniform). The remaining items were used up in training during the 1960’s (source: https://www.warrelics.eu/forum/bundeswehr-forum/bundeswehr-splittertarnmuster-camo-uniform-710411-2/#post1902467).
Jacket:
This jacket is known as the standard “infantry” version (source: https://www.wehrmacht-awards.com/forums/forum/ken-jasper-international-militaria-forums/bundesrepublik-deutschland-1949-present/694050-documenting-bw-splittertarn-variants). Any unit in the Bundeswehr authorized to wear camouflage could have used it (source: https://www.wehrmacht-awards.com/forums/forum/ken-jasper-international-militaria-forums/bundesrepublik-deutschland-1949-present/694050-documenting-bw-splittertarn-variants?p=5713516#post5713516).
The jacket is made of heavyweight cotton poplin material, similar to the most common versions of the heavy Belgian Moon and Balls and Brushstroke paratrooper jackets. The same material was also used for some Bundesgrenzschutz (BGS) jackets and the some French TAP smocks like these 47/54 and 47/56 examples. The cut is also similar to other western European countries’ paratrooper smocks from the same era. All these similarities could be due to the fact West Germany ratified the 1952 treaty to establish the European Defense Community (EDC), an early attempt at European integration that led to the modern (as of 2020) European Union, and a precursor to NATO. Although the EDC was not formed, the experience gained from designing uniforms for it would have been useful for all those nations in the 1950’s (source: https://iacmc.forumotion.com/t640-west-german-1955-leibermuster).
Tag:
November 1956 date
Inside Features:
Pass through armpit ventilation slit. 
Both sides of the chest have an internal pocket. The right one (as worn) has the tag.
Waist area has elastic on the inside to prevent it from riding up as well as to help retain heat, something other countries used a beaver tail/crotchflap for. The cut is very complicated and it used a lot of high quality metal hardware which would have made it very expensive to manufacture.
Pockets:
Metal D-ring was for a Swiss style backpack. meant for the EDC uniform, to be attached (source: https://www.warrelics.eu/forum/bundeswehr-forum/bundeswehr-splittertarnmuster-camo-uniform-710411-2/#post1903807). The jacket itself was meant to be load bearing gear for supplies and ammunition in magazines (source: https://www.warrelics.eu/forum/bundeswehr-forum/bundeswehr-splittertarnmuster-camo-uniform-710411-2/#post1903479).
Collar Fastener and Epaulettes:
Epaulette system 
Collar fastener.
Button is sewn to the jacket’s body and the epaulette straps can be attached in either order.
Cuffs:
Hardware:
Pattern and material is similar to the BGS Zeltbahn, but bluer and with more gaps between the shapes.
Usage Photos:
Bundeswehr:
All photos from this thread: https://iacmc.forumotion.com/t13008-splittertarnmuster-jacket-11-56 (warning: some BGS photos mixed in; I have selected the ones I can confirm have the same camouflage pattern shown on this page).
US Army LRRP:
V Corps Long Range Reconnaissance Patrol (LRRP) in Germany (from https://www.usmilitariaforum.com/forums/index.php?/topic/254421-v-and-vii-corps-lrrp-units-in-the-1960s/#comment-2791837). 
From https://eaglehorse.org/home_station/hidden_stories/60s/lrrps/3id_lrrps.htm.
Besides the Bundeswehr, items obtained secondhand were used in OPFOR roles by a variety of western countries (source: https://iacmc.forumotion.com/t8937-documentation-of-splittertarn-uniform-variants#77416) and by US Long Range Reconnaissance Patrols (LRRP) units stationed in Germany since there was no US standard issue camouflage uniform during that time period, and it would have helped disguise their American origin (source: https://eaglehorse.org/home_station/hidden_stories/60s/lrrps/3id_lrrps.htm).
The Beatles:
Other than military use, the jacket was famously worn by John Lennon of the Beatles in the Mad Day Out photos (1968).
Call of Duty Black Ops Cold War:
One of Ingo Beck’s skins (Commander) is of him “undercover” wearing the West German Bundeswehr splinter uniform.