While the law seems to imply that you are convicted for being a murderer at all, the actual wording is better translated as following. Emphasis is mine.
Murder under specific aggravating circumstances [Mord]
(1) Whosoever commits murder [altnernate: "A murderer..."] under the conditions of this provision shall be liable to imprisonment for life.
(2) A murderer [=someone committing murder] under this provision is any person who kills a person for pleasure, for sexual gratification, out of greed or otherwise base motives, by stealth or cruelly or by means that pose a danger to the public or in order to facilitate or to cover up another offence.
(1) Whosoever kills a person without being a murderer under section 211 shall be convicted of murder and be liable to imprisonment of not less than five years.
(2) In especially serious cases the penalty shall be imprisonment for life.
Murder under mitigating circumstances [Minder schwerer Fall des Totschlags]
If the murderer (under section 212) [=person committing murder] was provoked to rage by maltreatment in icted on him or a relative, or was seriously insulted by the victim and immediately lost self-control and committed the offence, or in the event of an otherwise less serious case, the penalty shall be imprisonment from one to ten years.
The actual wording is not "The murderer is punished with livelong imprisonment" but "committing the act of murder is punished with lifelong imprisonment. To commit murder in this way this has to be true...".
In the US, §211 would be basically analogous to "First degree murder" and "Second degree murder" together, §212 to voluntary manslaughter and §213 to involuntary manslaughter, defining variants of the crime, all relying on the definition that a murderer is a person that did commit the act of killing somebody as defined in §211(2)/§212(1) without the aggravating circumstances neccessary to fulfill §211(2). Not being a murderer is punished but committing the act of murder which makes the killer the murderer of the victim.
Because of this, let's design three cases:
Jonas Schmidt (~John Smith) loves to drive fast. He drives Janina Reh (~Jane Doe) over and she dies. He didn't commit "Mord" but "Totschlag" under §212, goes to jail for 5 years, then comes out... and does it again, driving over another person. Once more he goes into prison, comes out some years later and could arguably do another act of Totschlag! Though it's likely he'll lose his driver's license.
Kain (~Cain) hates his brother Able after his bonfire had the smoke rise to the sky while his made a huge cloud on earth. So he goes and murders him in a back alley, gets caught and gets convicted for murder under §211 because hating somebody is a base motive. After average 18.6 years he leaves prison and has could go out his way and slay somebody else for having a better bonfire building skill, though this time the judge might want to lock him up for good by mandating "Sicherheitsverwahrung" (the meaning of this is "lockup after the term is served for the safety of society").
By the way, there are cases in Germany where a released murderer murdered again. 45 of these were collected in the book Mord im Rückfall: 45 Fallgeschichten über das Töten by Hans-Ludwig Kröber.