It's common for people to think Tidus travelled through time because he spends most of the game convinced that he did. You could sum up Tidus' reasoning as follows:

1) I'm from Zanarkand.

2) The Zanarkand ruins in Spira was destroyed 1000 years ago.

3) Therefore, I'm from the Zanarkand ruins and I traveled 1000 years in time.

However, his logic was faulty since he assumed his Zanarkand had to exist in the same place as the Zanarkand ruins. Since the ruins have been there for 1000 year time travel seemed to be the only way to explain it. The ruins of Zanarkand are clearly not DZ though since it's ringed by mountains and DZ is on an island completely surrounded by water.

So DZ exists in a separate location in Spira. It coexists in time with the present-day ruins but not in space. That permited Tidus to simply travel from DZ to elsewhere in Spira via Sin.

As to the appearances, the past view of Zanarkand that Seymour shows to the group in Guadosalam does look tecnologically advanced as present-day DZ. Since Spira was at it's technological peak 1000 years ago this is not suprising. However, the rest of Spira reverted to a less technological state while DZ maintained the pre-Sin technology levels.

So in summary, DZ coexists with the rest of Spira (including Zanarkand ruins), but completely isolated. Shuyin is from the old pre-Sin Zanarkand of 1000 years ago while Tidus is from present-day DZ.