Yes, aññā is from the same root, but has a distinct meaning. Saññā means “perception, recognition” while aññā means “enlightenment”.
The Indo European root “to know” (g̑en), found in Pali as the various forms of the root ñā, is one of the most prolific and characteristic of all PIE roots. Indeed, so much so that it is still extremely well represented in English. It’s why we have a “k” at the beginning of “know”; compare Sanskrit jnā with Pali ñā. It’s also found in roots like “ken”, “cognize”, etc. The PIE root u̯(e)id- (Pali vijjā, vijjati, veda, vedanā, etc.) is also strongly represented in English as “wit”, “witness”, “witch”, “wicca”, etc.
I think this is very significant, like cows and horses
Why? Because for both of these we use the PIE root in Indic and European languages: gu̯ou/cow/go and * ḱr̥sos/horse/assa.
And this is for a very well-defined historical reason: horses and cows were extremely important to the ancient PIE people, and in fact they probably domesticated the horse (and the cow? I don’t know.) When they traveled to Europe and India, the things that were crucially important to them made a lasting impact on the people of their new lands, and remain in the language today. There’s no PIE root for, say, elephant, because they weren’t found in the PIE homeland.
So I think the very idea of “knowledge” was an Indo-European characteristic or invention. Not that other peoples don’t have knowledge, but perhaps it wasn’t as central, as reflected on, as definitive a part of their life. When the PIE people traveled the world, they didn’t just take their horses and their cows, they took their “knowledge”.
And this gets even more speculative, but I wonder if the two things are related: that is, is the very fact of the portability of PIE culture related to their idea of knowledge? Normally, knowledge is passed down in an embedded way in culture; it is fixed to the landscape, the people, the village, the stories of the elders. But when you go to another land, this traditional embodied knowledge or “kenning” is disrupted. You need to think about what it is that you know, how it is that you know, and how that knowledge can be preserved in a portable form, i.e. memory.
Could it be that the diaspora of the PIE peoples, for reasons we can only guess at, was the initial spur behind the creation of knowledge and learning as we know it today?