Translation:He thought that I was an architect.
Shouldn't the correct translation be "He thought that I WERE an architect"? A past subjunctive form of the verb "To Be" is used to make a sentence sound more formal, right?
No. We simply don't use were in statements like this. You need a condition like if to use were, and the sentence would then seem incomplete - He thought that if I were an architect...? If you used were in this sentence, you would sound like you were speaking a non-standard dialect. (The subjunctive, such as it is, is fairly rapidly disappearing from English, but might be found used more in a place like the Appalachians in the Eastern part of the US..)
The subjunctive form, as opposed to the indicative form, has many uses and certainly does not restrict itself only to sentences with "if". I'd say that "were" in this case could be perfectly correct.
As you both indicate, the subjunctive form in English is fading away and sounds rather formal these days. So the indicative form "was" is of course always fine too.
Some more on the subjunctive form:
HM. Having the subjunctive in this sentence if it stands alone and is without the if HAS pretty much disappeared completely from English. He thought that I were an architect is just plain wrong. One of the problems with the English subjunctive, of course, is that it sort of gets buried in past tense and conditional as well, the forms are pretty much the same, so it's sort of hard to sort out when which is what.