Answer:
No, the bassoon's fingerings are not the same as the fingering of oboes.
There was a time, back when oboes were first invented and bassoons were just changing over from the single-piece dulcian to the four-piece basson/bassoon that the fingering of the two instruments was quite similar, and had much in common with the recorder and flute as well.
With the split between the German and french schools of double reed instruments, though, it's hard to say that all bassoon fingerings are alike, or that all oboe fingerings are alike!
The major difference in fingering between all oboes and all bassoons comes from the fact that the bassoon's bore is folded. This means that two lengths of the bore, the part that makes high notes and the part that makes low notes, run parallel to each other. As a result, keys that affect low notes appear near keys which affect high notes, and fingers serve multiple duties.
Because of the size of the bassoon, it is generally supported from the lower end by a belt-like strap which the player sits upon. The oboe, though, is smaller, and is generally supported by the right thumb of the player. The immobilized thumb of the oboist and the free right thumb of the bassoonist lead to further differences between the instruments. The oboist has multiple keys which are operated by the little fingers and a key and a hole with a ring (the ring acts like another key) under the left thumb. The bassoonist, though, has two keys under the left little finger, as many as four under the right little finger, between 3 and 5 under the right thumb, and anywhere between 8 and 12 keys to operate with the left thumb!
If you were to consider the basic octave, though, there is a strong similarity between all of these instruments: the three fingers of the left hand (minus the little finger), the four of the right hand, and the left thumb are employed in playing the basic octave and most of the next octave. However, in all varieties of oboe and bassoon, accidental keys, whisper/voice keys (which help to sound upper octaves) and lower-range extension keys differ greatly.
I play the oboe and tried the bassoon for a few weeks. I found that the fingerings are the more or less the same but at a fifth difference, so an F on the oboe is a Bb on the bassoon.