I've just run into the same issue. I smithed both the top and base of the iron anvil separately, kept them heated, and followed the steps to create (place base, shift-click powdered borax, place top, hammer). I just keeps hammering the top lower and lower into the base, probably ~30 times before I finally gave up and broke it, reclaiming both pieces of the anvil and the borax.
I saved & quit, reloaded the world, and tried again, and it worked as expected. I broke the iron anvil a couple times to test if it would break as a complete anvil, and it did just fine.
So it appears simple exiting & reloading the world after crafting the top & base and *then* attempting to assemble should get around the bug.