Here is one reason: a note with a fundamental frequency of 100 Hz will have harmonics at 100 HZ, 200 Hz, 300 Hz, 400 Hz, 500 Hz, 600 Hz, etc., while a fundamental of 200 Hz has harmonics of 200 Hz, 400 Hz, 600 Hz, etc. These are a subset of the harmonics of the 100 Hz note an octave below. The human auditory system detects the pitch of the fundamental largely by inferring it from the harmonics. You can test this by playing sine waves at, say, 400, 500, 600, 700 and 800 Hz simultaneously - you will hear it as a note of 100 Hz, even though the fundamental is not really there. Physically, the shape of the cochlea does something very like a Fourier transform on the incoming sound. If you "unroll" it, the cochlea is effectively a long tube of slowly varying width, and the width of each part of the tube determines its resonant frequency.
The upshot of this is that one reason for the auditory similarity between a note and its octave is due to the fact that they share so many harmonics. If you play a note at 100 Hz and then simultaneously start playing a note at 200 Hz, it makes some of the original note's harmonics louder but doesn't introduce any new ones.
Of course, the same would be true with if the new note had a fundamental of 300 Hz rather than 200. However, in this case, while the high note still shares all its harmonics with the low one, but the low one only shares a third of its harmonics with the high one. Perhaps this is why we perceive the octave as a more consonant interval.
This can also help to explain the consonance of other intervals. For instance, the harmonics of a note at 150 Hz (a perfect fifth above 100 Hz) are 150, 300, 450, 600, 750, 900 Hz, etc. You can see that it has a lot of harmonics in common with a 100 Hz note, and this is part of the reason we find these two notes consonant and, in some ways, quite similar-sounding to one another. But there are less harmonics in common between a 100 Hz fundamental and a 150 Hz one than there are between a 100 Hz and a 200 Hz fundamental, which perhaps is why the octave sounds so much more consonant than the fifth.
It's worth noting that automatic pitch-detection algorithms also find octaves (and fifths) "similar" to one another, in the sense that one of their most common errors is to mis-classify a note as one an octave above, effectively because they fail to notice the even harmonics.
Another interesting thing to note is that grand pianos are tuned to a scale that repeats at an interval slightly greater than an octave. This is because the heavy strings don't quite obey the ideal string law, and have harmonics that are slightly "stretched". So if you want the piano to sound in tune, you have to stretch the octave slightly as well.
However, despite this physical basis for the equivalence of the octave, there's a big cultural element in it as well. Not all musical traditions place the same emphasis on consonance as western music, and not all human cultures consider notes an octave apart as being equivalent to one another. Some composers have even experimented with scales in which non-octave intervals (such as an octave plus a perfect fifth, i.e. a factor of 3 rather than 2) play the same role that the octave usually does.