Google Trends and Google Analytics are not the same thing. Google Analytics is not relevant here (it can't be used for this). People are using Google Trends, which is designed to show how the popularity of common search terms varies over time. To my knowledge (MOO, possibly out of date information disclaimers apply), it operates on a sample of search data rather than all data. For example (completely making up the number here), they might decide that they only need to look at 1% of search data to generate trends, because any sufficiently popular search term will still be in those millions of searches, in the right proportions compared to other popular search terms. But if you have an uncommon search term, it's possible or even likely that it won't show up at all in that 1% slice. And maybe it does show in the next 1% slice and it looks like suddenly there's a spike in traffic, when it was roughly the same.
I would not expect the problem to be that it shows searches that didn't happen. I expect that it will miss searches that did happen, but potentially inconsistently, which can show up as spikes and valleys that don't actually exist.
It's also possible that some search term normalization happens (things like spell correction, clustering of similar search terms, etc.) which might cause searches to show up in a trend when you might not expect them to.
Finally, the geolocation stuff is really iffy at small numbers. If all of the searches in that "spike" come from one user who happens to be using a VPN, it's going to be way wrong. This effect generally will cancel out on a common search term with millions of searches.
Bottom line really, without needing to know any technical details of how this works, is that it's not what Google Trends was designed for. Just like any other time we're using something in a way other than what it was designed for, the results might not be what we hope.
JMO, based on a combination of experience, knowledge, and educated guesses. There may be mistakes here. Trust at your own risk.