My closest matches are "extended family". A second cousin once removed with whom I share 177 cM and the link is through my great grandparents on my mum's side.
DNA does not lie. if you are matching with someone, there is a DNA link (leaving aside the issue with random mutations of DNA and matching with only a dozen of centimorgans)
Ancestry suggests the relationships based on your DNA numbers, the people in your family tree if you have built one on the site, and hte people in their tree. "Jane SMITH could be your third cousin through your great great grandparents John JONES and Mary BROWN" and so on.
But it's not exact and the suggestions can be wrong, especially if one person has not completed a tree. We have 4 pairs of great grandparents, 8 gg grandparents and so on. And half-relationships complicate matters even further, if an ancestor has remarried, or had an affair or something.
One thing I think it's important to mention - although I don't think for one second that it is relevant to the OPs case. Everyone likes to joke about the milkman, or "love children" or affairs and that's often the case. But not always. Unknown parentage can be caused in more sinister ways too. Many people forget that before the 60s, termination of pregnancy was illegal in the UK. Women/girls were forced to have their babies and often gave them up for adoption if they were single, whether raped, abused, or just pregnant by their boyfriends.
There have been so many stories in the press about women now quite elderly who are absolutely devastated by what happened to them and who have never come to terms with it. You never know if you do DNA what you might discover not just affecting your close family, but also aunts, uncles, cousins. All families have secrets.
It's SO important to tread with caution and although it might be really difficult, accept that people have the right not to engage with you. Not immediately leap in half-arsed with little understanding and cause huge upset as this person did to the OP yesterday.