The Kindle Fire HD 7" will have a worse SOC than the Nexus 7
The Kindle Fire HD 7" will have the same SOC from TI as the Galaxy Nexus, According to Anandtech: http://www.anandtech.com/show/6246/amazon-kindle-preview-paperwhite-fire-2-and-fire-hd-7-89.
This means that the Fire HD 8.9" and Fire HD 7" will have two different System on a chip. The larger Kindle will have an OMAP4470 which probably is a little bit better or equal to the Tegra 3 chipset. The smaller Fire will have the OMAP4460 which is worse than the Tegra 3 in most benchmarks.
So the take away is that the Kindle Fire HD 7" will have a chipset which is worse than the Nexus 7.