Technically, the manufacturer probably could have used a lens with a larger diameter, and thus a larger maximum aperture. Only they could answer definitively why they didn't, but it could be a result of any number of things, e.g.:

1. Larger diameter lens would also be thicker to maintain same focal length, which has implications on placing the lens at the right distance from the sensor, how far the lens extrudes from the body of the phone, and thus the design of the phone case itself, and also uses more real estate on the inside of the phone, which means other components may need to be moved around or reduced in size
2. Larger lens is probably more expensive to manufacture
3. Larger lens may be more difficult to manufacture to the same standards with regard to distortion and aberrations

That's just a few off the top of my head. There's probably quite a bit more. It comes down to design choices.

Oh, and with regard to your opening question - there isn't really any relationship between maximum aperture size and sensor size. Aperture is a characteristic of the lens - in the example of cameras with interchangeable lenses, you could take the same lens with the same focal length and aperture, and attach it to different camera bodies with different sensor sizes. If sensor size and aperture were somehow dependent, that would be more difficult...