Good Evening Lee,
I'll happily give my thoughts on Stoke City and memories that I have, albeit as Metty says above, we haven't played for 16 years so it might be fairly sketchy but I will give it a go. I'm only 26 so my memories are from when I was in Primary and Second education. However, I'm sure my dad @unsworth blue will be on with his memories of Stoke City.
1. I certainly don't see Stoke City as a rival. Despite only being just over an hour away geographically, there are so many teams that are around us that anything further than 25 miles away is not really considered a rival. That as well as not playing Stoke very often as previously mentioned. My personal thoughts of Stoke is that they have been and are well supported but certain season's of Pulis football lost any potential of describing Stoke as a lovely footballing side (sorry!)
2. Growing up, I used to see Stoke teams playing against Latics which included the grey hair of Graham Kavanagh in midfield with Peter Thorne up front and latterly Chris Iwelumo (don't ask Latics fans what their opinions are of him, I'd guess it's not great amongst the majority from his spell here). Stoke when I were young, always seemed to be up near the top of Division Two (League One now)
3. The games that stick in my mind when growing up, against Stoke City was a Tuesday night game in 1998/99 (my first season I can remember) and needing 6 points in the last week of the season, firstly against yourselves, then against Reading. We managed to beat Stoke 1-0 and Reading 2-0 to secure survival in League One. The other game that sticks in my mind was Stoke winning 2-1 at Boundary Park back in May 2001. That sticks in my mind purely due to the Oldham riots in which Stoke City fans were involved both before and after the match.
4. I always found the Britannia Stadium quite loud and as such the following support being loud too, especially at the start of the "delilah" song. I have always found that Stoke have travelled in decent numbers however that being said, they have mostly been doing pretty well when playing at Boundary Park.
5. I'm not old enough to have visited previous home grounds of Stoke but the Britannia is nicely built albeit bloody freezing when the corners were open. That said, playing at Port Vale year in year out, it always seems cold there so maybe that part of the country. However, sounds pretty rich when Boundary Park is exceptionally cold during the winter months and at times during the summer.
I hope this is an insight that may be helpful to you and best of luck in your quest. I should imagine, due to reputation and stories told to me by older Latics supporters that feedback might be mixed but all the best.