From an amateur student of these things, I would say its most likely a test-cutting on a cadaver. In this case, two cadavers, one stacked on top of the other, as was typical for test-cuttings.
All of this assumes the inscription is genuine. Bear in mind that there is a huge, huge industry and trade in forged swords, and swords coming from overseas with suspicious inscriptions on them. I looked in my references for any information on Nobumori, but I couldn't find anything. So its a bit mysterious.
Tameshimei are not so uncommon. Some purists don't fancy swords with this kind of inscription. Usually one wants to look at the sword itself and not at the inscription. But other collectors find them very interesting.
Thank you for the extra information. This is fascinating. Pretty amazing to find this on the tang of a sword.
This is the other side of the tang. Would anyone know what it also says?
Thanks again for all of the help.
This is the business side of the tang. This side has the info that helps determine the sword's value. The test-cutting is interesting, but the swordsmith himself is more important. But what you really want to be looking at and studying is the sword blade. The part that gets hidden inside the pommel is window dressing (albeit, important window dressing in some cases). But if you are thinking of buying this sword, don't be too dazzled by the inscription on the tang. It is like being offered a Rembrandt painting and getting sidetracked by the frame. There might be rare cases where a well-preserved contemporary frame will help verify the painting's authenticity, but it would never be more interesting than the painting itself.
Takakiyo is as much of a mystery to me as the test-cutter on the opposite side. Takakiyo doesn't show up in any of my references either. Its usually not a good sign, but if the sword itself is forged well, one could overlook the very obscure smith and test-cutter. If somebody is trying to sell the sword to you for a high price, and trying to justify it by the "unique" inscription, I would proceed with extreme caution.
A good resource is the Nihonto Message Board. Full of amazingly knowledgeable people who would be keen to help and advise.