Unfortunately I have to disappoint you regarding the whole story of this soldier for multiple reasons:
1) Germany takes data privacy
very seriously. If you want to research people, you have to have a valid reason and sent certain filled out forms back to the state military archives. I.e.: if you research your own family members, or in case you do research for a scientific paper intended to be published, which has info about this person,etc. There are some more options you can choose from which
valid reason suits your needs best, but having random "Max Mustermann " looking up the unrelated (dead) stranger "Hans Müller" on the internet is a big no-go here in Germany.
2) Even if such websites like the US ones exist (type in the name, and receive the whole military files including vaccinations/STDs/etc.), you'll never receive the whole picture where someone served from beginning to end. During WW2 the military archives in Potsdam near Berlin were bombed out completely, so only fragments survive of what was once an enormous library of military personnal info reaching all the way back to the 30-years-war in the 1600s.
When I inquired about my Great-Grandparents, I only received excerpts from various field hospital medical records which showed his unit
at that point in time in the field hospital. It's basically logbook style: when he got in, what his health problems/injuries at that time were, when he left and to which unit he was assigned back. But It doesn't tell me when he joined/was transferred to a certain unit during the war. These excerpts are only "snapshots" in time. For other ancestors, I didn't get lucky and the archives didn't find anything about the person at all.
But there are certain "free-to-use" tools, like searching in the "Gefallenenlisten" (Lists of fallen/injured soldiers)
https://des.genealogy.net/eingabe-verlustlisten/search or inquire at the "Volksbund" for war graves in case the soldier was KIA and buried
https://www.volksbund.de/erinnern-gedenken/graebersuche-online (requires registration). The first one are digitalised lists of soldiers that were publicly displayed back in WW1 and mostly showed only name, DOB (only Day & Month) and KIA/WIA status, as to not inform potential spies about military intel.
So the only "hard" info you'll probably ever have in regards to his military service and unit(s) is the one on the card at that point in time:
Otto Marquardt, Maler ["Painter", probably for houses."Maler" can both mean "artist painter" and "gives houses a paint coat" in german]
Soldat im 3. Batl. Res.Inf.Rgt. No.244
53. Division, 27. Armeekorps, 9. Kompagnie
am 9.10.1914 [for my american friends: DD/MM/YY, not MM/DD/YY
]
Leipzig-Kleinzschochau
Dieskaustr. No. 103 I [his address with apartment No. in Leipzig]