Can I connect a second keyless light fixture and control both from one pull-chain?
Possible, with a change of fixture

I mean, we are dealing with a 2 dollar lightbulb holder here.  

The problem is, we will need to feed the downline light with a) neutral, and b) switched-hot: the electrical point between the switch and the bulb.  

On this particular fixture, the switch and socket base are integral, and they don't make that point accessible. At all.   

So you'd need to find a model of fixture where they do expose it at a tappable screw.  

At that point, neutral would stay where it is, and hot for the downline cable would be placed on the screw.