The Wikipedia article about Concorde confirms that the name is typically used without an indefinite or definite article, at least in British English:
Concorde also acquired an unusual nomenclature for an aircraft. In common usage in the United Kingdom, the type is known as "Concorde" without an article, rather than "the Concorde" or "a Concorde".
The two sources given for this sentence (BBC and British Airways) also consistently use Concorde without an article, but unfortunately they don't explain why.
I remember hearing in a documentary that the British developers omitted the article to highlight how special the aircraft was, but I cannot find a source for that. The closest I found is this Engineering and Technology article:
Unlike almost every other commercial airliner, Concorde sparked such affection that ‘she’ needed no preceding definite article or proceeding marquee number.
So the reason is probably related to how special the aircraft was and how affectionately the people working on the project felt about her.
By the way, in French Concorde is used with an article: le Concorde (see e.g. the French Wikipedia article).