Stormfather seems like an oddity to me as well. Other cognitive shadows are... themselves, right? Kelsier is our best example, and he doesn't become "attached" to some kind of cognitive entity when he died.

I have two guesses... (with perhaps a bit of a blurred line between them)

First, a more crude explanation, is that Tanavast's cognitive shadow became "attached" to Stormfather somehow after being created. So Stormfather (the spren) is kind of like the body we see in most other cases. A dead man is given a Divine Breath and then re-attached to his body. Kelsier hung on using Investiture from the Well, and then later on seems to have re-attached himself to a/his body--probably using hemalurgy. So maybe, for whatever reason, Tanavast's cognitive shadow stapled himself to Stormfather.

Second, we know that Honor was Invested in Roshar's spren, right? It makes sense that he would have been particularly Invested in Stormfather, given the "size" of the spren. So I think it stands to reason that, when Tanavast died, the cognitive shadow itself was made out of Stormfather's own Investiture. Hard to explain... When Kelsier died, he had to run to the Well to "soak up" enough Investiture to allow him to stay permanently. The idea here is that Tanavast already had this "well" of Investiture at hand, in the form of Stormfather. So Stormfather is basically serving as the same thing as the Well of Ascension did for Kelsier. The oddity is not so much that Tanavast's cognitive shadow became attached to a spren, but that Honor was so heavily invested in a sentient cognitive being (a spren) to begin with.