I’m not sure if it is real hypercorrection (or even attempted correction) or a conscious adoption of an available model pronunciation of plural nouns with similar spelling as a tool to distinguish the plural noun from the verb.

Though, in either case, it has the danger of suggesting the plural form of the hypothetical word “processee” (from processor, and by analogy to pairs like lessor/lessee), so, “things that are processed”.

The “hypercorrection” being to use the same long plural e from axis/axes with process/processes?

> American pronunciation

..since when? I don't think I've ever heard "processees" before...

I've heard both used in the US.