If another Spirit gets hold of Elemental Teachings (from Shifting Memory of Ages), can they Prepare element markers?

If another Spirit gets hold of Elemental Teachings (from Shifting Memory of Ages), can they Prepare element markers?

Yes. While Shifting Memory's panel defines what "Prepare" means, other Spirits can do so if they are able to use a Power that instructs them to. They can spend those element markers on Elemental Teachings or for whatever Shifting Memory's special rules say those markers are for (normally getting extra Elements for a single action via Insights Into the World's Nature).
While a strict reading of the text of Insights Into the World's Nature would have the other Spirit store its prepared markers on Shifting Memory's panel, don't do that. It just wasn't worth convoluting the wording of that paragraph to cover the extreme edge-case. As of the release of Nature Incarnate, "Prepare" is a general game keyword, so the exact text of the first paragraph of Insights Into the World's Nature is just a reminder of how things normally work for Shifting Memory of Ages.
If an Aspect replaces Insights Into the World's Nature, the function of Element markers also changes for other Spirits. For example, if Shifting Memory is using the Intensify Aspect, Element markers can be spent by other Spirits to increase the effects of one of their Spirit Actions, but can't be spent to gain Elements for a single action.