I've found a bunch of other different combinations of microphone sharing between the Beatles. Here's George and John:

Here's Paul and John:

Here's George, Paul, and John:

Here's Paul and Ringo (though obviously this is a little more recent):

A Google Image search of "Beatles sharing microphones" doesn't seem to turn up photos of George and Paul any more frequently than John and Paul, though photos of all three and of George and John are both less frequent.
I also found a few forum posts asking the same question, and a variety of potential answers (along with bad puns and off-topic arguing, because, you know, they're forums) was offered, including:
- It made it easier for the two back up singers to harmonize.
- On 8 track recorders, there would be 3 tracks for the guitars and 3 tracks for kit, which would leave two tracks for the vocals.
- It looks cool!
- Early Beatles songs usually had only John or Paul singing lead, so George didn't need a microphone.
According to Ringo Starr, it's the first one:
John and Paul always sang into the same microphone when they did harmonies, and if George was singing he'd be on the same mic too. And that’s why the harmonies are so great because they’re close to each other and they can hear each other.
I should note that Ringo could be talking about singing in the studio, rather than live. However, a lot of well known artists implement/have implemented microphone sharing when performing live, whether occasionally or frequently, like: