*Assassination, public execution still implies a degree of polelitical legitimacy; alternatively, the assassins misinterpreted the meaning of polish remover when doing their nails
There's a historical/cultural reason behind such choice of words.
During the WWII the Polish Underground State (i.e. the resistance) maintained continuity of political and governmental structures of pre-war Poland, including the judicial branch with the so-called Special Courts. These were intended for prosecution of common criminals (since the Germans were a bit slapdash about that), collaborators, and occasionally occupants.
This of course meant that the defendant was more often than not judged and sentenced in absentia, before being 'executed' when the opportunity arises.
That is to say, from the point of view of the occupying Germans, these were assassinations by partisans. From the point of view of Poles, these were executions by the Polish state.
The All-Polish Youth and the likes of them (basically, our version of alt-right) want to paint themselves as a structure similar to the WWII underground, currently being suppressed by their bogeyman (occupant) of progressives, the EU, and whatnot. One of the easiest ways to spot a likely alt-righter is by their apparel - they do love their WWII-themed patriotic clothing (google 'Polska walcząca bluza' for examples).
Anyway, threatening with 'execution' is meant to reinforce this association with the legitimate patriotic government under occupation by basically Nazis. Any Pole who hasn't completely flatlined in school would pick up on that.