My dough won't rise the second time
I can't say what you did wrong or if it was the yeast at fault as suggested as we can only go by your results. I'm not so sure the yeast would be faulty though since you said it doubled in the first rising.

They look like they 'sort of' rose but more sideways since they're melded against one another. Or did you put them touching each other to begin with? The baked buns in the link have a wide spaces between them so if you spaced them that way, the did rise but not properly. 

If you left spaces between them and they spread out when rising but didn't hold their shape, how soft was your dough? Softer dough with higher hydration don't rise high and keep their shape as well as lower hydration doughs.

If that doesn't fit your dough, is it possible that the temperature used in your first rising was too warm? A couple of times when I was trying to hurry the process (even though I know better), I've subjected the dough to too much heat. It rose quickly and doubled in bulk but most of the yeast was killed. The second rising didn't go well with most of the yeast dead. Is it possible you did something similar? 

Sorry to ask questions instead of giving a concrete answer but all we can do, from our end, is guess and hope something fits. Looking carefully at your photo, there are air pockets like bread should have but with a mix of small and big spaces like dough with high hydration has. The surface of the buns are uneven too. When you shaped the buns for the second rising, we're they rounded and smooth like the unbaked buns in the link? 

Hopefully if you provide more information, others may see other possible reasons.