"I know that this would be very very small connection and pretty much insignificant but it still fascinates me regardless!"
This can happen quite a bit.
"to add I have one ‘both sides’ match on ancestry"
I have my DNA on Ancestry and both my parents also have their DNA on Ancestry.
I have 23 matches that are "Both sides". But, at first, some of the information seemed a bit odd.
For example, I have a number of "Both sides" matches that only show up under one parent but not both. Even more strangely, I generally share more DNA with that person than my parent does.
I asked Ancestry about that and this is how they explained it. Ancestry only record matches above a certain level of shared DNA (I think it's something like 8 or 9 cM for a direct match or 20 cM for a shared match).
In one case, I shared 17 cM with a "Both sides" match. But my father only shared 11 cM with that match and my mum did not show this person as a match at all.
Ancestry explained that my mum would have shared less than 8cM with that match and so it wasn't recorded. And, between my dad's 11 cM and my mum's less than 8 cM I inherited 17 cM from both of them and so that is why this person shows as a "Both sides" match with more shared DNA.
Or, slightly closer to home, I have two half 2nd cousins through my mum (same great grandfather, different great grandmothers) that I share 302 cM and 201 cM. They are 2nd cousins on my mum's side but they are also 8th cousins through my dad who is distantly related to the other great grandmother that my great grandfather married secondly.
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My dad has 12 matches that are "Both sides" and my mum has 11 "Both sides" matches.
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"I have a lot of matches that are labelled maternal or paternal only but share matches on both sides, particularly really distant matches"
In some parts of the world is was not uncommon for the same families to intermarry over the generations. This was especially true in areas that were isolated or where there was a difference of religion or language for example.
This means that you are much more likely to have DNA connections through both sides of the family. This also means that the same DNA is kept within the family for a longer period of time than you would expect normally so the connections may be further back than you would normally expect. There's even a name for this, it is referred to as "endogamy"
Areas like Cornwall (and Norfolk) were prone to this. The Isles of Scilly are a great example; back in the 19th century it is estimated that around 80% of the marriages there were endogamous to Scilly. In other words, they just married each other.
But this also happened in other areas as well. I have some distant ancestors who went over to the American colonies in the 1600s. To be frank, there were a lot of families intermarrying each other back then (the population in Maryland and Virginia was very small) and you can still see the results of that in the DNA today.
Many Americans who trace their ancestry from early US colonial families have found this when getting their DNA results.
I remember somebody saying that they had colonial ancestry from Maryland and Virginia through both parents. While the parents weren't related to each other in any way over the last two hundred years, they had dozens of DNA matches who were related to them through both parents, and quite a few more who had a DNA match to one parent and a paper trail that connects them to the other.
It is not uncommon for colonial Americans, Ashkenazi Jews, French Canadians and US Cajuns to still show the results of this endogamy from centuries earlier.