Is that bizarre conspiracy theory more likely or that their search results scrape Wikipedia and return the most recent party a politician belonged to?
https://en.wikipedia.org/wiki/Abraham_Lincoln
Lo and behold, it reports that Strom Thurmond was a Republican (https://en.wikipedia.org/wiki/Strom_Thurmond) and Jeff Van Drew is too (https://en.wikipedia.org/wiki/Jeff_Van_Drew), even though both were Democrats for substantial periods in the past.
Google for: "what party was Andrew Jackson in"
For me this shows "Democratic-Republican Party" - this was the first party he was part of - he was elected president while a member of the Democratic party and remained such until his death: https://en.wikipedia.org/wiki/Andrew_Jackson
So your theory does not hold.
Here we have someone who was a Republican being portrayed as something else and someone who was a Democrat being portrayed as something else - and in both cases there is political benefit to Democratic party in the US.