I can't answer the question about the dna process, but 4 x great grandparents is a whole lot of room for error somewhere along the line - it's very possible that one of you has mistaken someone with the same name/dob for a relative, or that a relative in those many generations didn't have the parents it says they did.

(Speaking from experience here going both ways - we have a beautifully detailed and complete family tree going back many, many generations but once you get back to the great great greats it's actually someone else's family tree for the simple reason that if you check the records carefully our relative was born 2 years after his father died..)

It's possible that you have inherited DNA from your ancestors that she hasn't - each time you inherit half from each parent, but you could be brother and sister and both inherit the different halves, and therefore have none the same (in theory).

This might account for it.

5th gen is very distant.
I often find dna matches who don't match my parents and match me or vice versa at that level!

This is really normal. If you think about how DNA works... you get 50% from your mum and 50% from your dad. A full sibling does too - but not necessarily the same 50%. It's like putting a pack of cards with your mum's DNA, and a pack with your dad's DNA, shuffling them, and cutting in half. Now repeat that process many, many times back through the generations and it is entirely likely that someone who is genealogically related to you has no DNA link to you.

isogg.org/wiki/Cousin_statistics

According to the International Society for Genetic Genealogy (ISOGG), less than a third of 5th cousins share enough DNA for it to be detected on an Ancestry test. Doesn't mean the test is dodgy, doesn't mean Ancestry is dodgy, or a scam, or inaccurate.

The same goes for exceptionally distant matches which you can't fit in. whoareyoumadeof.com/blog/can-dna-matches-be-false/ Once you get down to matches in the single centimorgans range it just means that a teeny tiny part of your DNA has mutated to match a similarly teeny tiny part of someone else's DNA spontaneously. I usually disregard matches under about 15-20 centimorgans - totally pointless.

Unlikely to be wrong, especially when you consider just how many people who have done their Ancestry DNA are American. If any of your distant relatives emigrated there a couple of hundred years ago, they and their descendants may very well have had children with people from other parts of the world. Their descendants will be the ones you are matching with.

My guess would be that someone, probably this other person, has got their family tree wrong. I'm very careful about who I add to the family tree and I've seen so many errors on Ancestry. It's actually rare to find a tree that hasn't got a mistake on it and some of them are jaw dropping. My money would be on that.