The shared ancestor hint is just that -- you match DNA with someone and you both have the same ancestor in your tree. Keep in mind they may, or may not, be the source of the DNA match.

The percentage is about how sure Ancestry is that your match based on DNA would be your 3rd cousin, but not that they have located a 3rd cousin. They might be a 2nd cousin or a 4th cousin or not a cousin at all.

The surname list just shows common surnames between your tree and your match tree.

It is all just hints for you to look at when investigating whether a match is actually related or not.

I am jealous about a 3rd cousin match -- my closest are only 4th cousins and none of them are 4th cousins. More like 6th and 7th and distant cousins! LOL!