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So, this is likely just the randomness of gene inheritance.
If we express cosanguinuity as percentages, you and your parent are at 50%, you and your grandparent 25%, etc. You get half your DNA from each parent, after all. But what about siblings? With siblings, you get into averages. You and your full sibling each got half your DNA from your mother and half from your father, but because the selection from each is random you could share anywhere from 0% to 100%. Rather than a flat 50%, you get a bell curve that peaks at 50%.
What if your sibling has a child with someone unrelated to you? Well, you and your niece or nephew are probably at about 25%, but because siblings are on a curve and there's a pair involved, you could be anywhere from 0% to 50%.
Similarly, first-cousins are typically about 12.5%, but 25% wouldn't be out of the realm of possibility and you could even get 50% if, say, their fathers are identical twins. If you and your cousin are simply on the upper end of the cosanguinuity bell curve, I could easily see one of those systems getting confused and thinking you're half-siblings, who would have a curve from 0% to 50% and peaking at 25%.
In short, testing just two random relatives doesn't actually tell you a lot unless you're testing a (supposed) ancestor and descendant. You would need to also get your parents and your cousin's parents tested to get anything definitive, and testing your grandparents too wouldn't be a bad idea for accuracy.
That also assumes any two unrelated people share 0% of their DNA. We all have common ancestors, if your uncle married a woman that happened to share 10% of your mom's DNA then you and your cousin could have more DNA in common as well.
Thank you, that seems a lot more likely than either one of us being secretly adopted by our aunt and uncle.