Right at the entrance a while interesting, but (I believe) unanswerable question.
Because “Vectrex” is an artificial name for a particular thing. Depending on, to what we expect in the thing, one has a male, female or neutral Vectrex: The game computer, game console, the device.
Even the omniscient Internet is no clear picture, as a search on Google shows:
“die Vectrex”: 979 hit
“the Vectrex”: 839 hit
“the Vectrex”: 806 hit
EDIT: The April results 2021 are a little clearer:
“die Vectrex”: 1710 hit
“the Vectrex”: 1220 hit
“the Vectrex”: 942 hit
The former manufacturer MB (Milton Bradley) drops the item off completely and writes only “VECTREX”. So it is, for example, on page 11 German manual “Important notes for the case, that Vectrex is not functioning properly.“. In the use of “Vectrex” as a name that's right, because you write, too “Peter goes home.” and rarely “The Peter goes home.”.
Personally, I only know “die Vectrex” and can not even tell, Though I initially was thinking of the game console. Who's is different usual, always with the articles here “the” or “of the” read. 😉
I also always tell the Vectrex. In addition, the Playstation, Xbox, die Dreamcast, the Wii, das NES, the SNES, the Atari 2600, the Gameboy, the Gamecube, Saturn, of / Atari Jaguar, the Master System, Megadrive, the N64. Probably because not all items correctly, some I now have to think longer. It means also the Amiga, although Amiga is in Spanish girlfriend. Somehow illogical.
Then we are already two, Axel. 🙂 Kind of funny with the names, but some names you come indeed to ponder: Andrea, Michele, Robin, Sascha, Ulli and some more. In the console I'm almost with you, But I say the Atari 2600, the Saturn, die Atari Jaguar (mentally with me all game consoles). But as it is written there is probably no right and no wrong. The Amiga's a funny contradiction, I had never noticed so.