In Pokemon Black & White there was the Vs. Seeker that let you have rematches with some trainers. Do X and Y have a similar item or mechanic?
There is no Vs. Seeker in XY, but there are a couple of places where you can rebattle trainers and get some experience, if that's what you're after:
Pre-ending of game
- Battle Chateau - On route 7 (You get new trainers every 5 minutes or so from my experience). Once you reach the highest and second highest ranks you can fight Gym Leaders and Elite Four members here as well. If you use Writs the Elite Four members may have Pokemon up to level 80. And you can use different writs to affect the level of the trainers' Pokemon and the number of trainers for a certain amount of time.
- Lumiose City - In alleyways provided you have already restored power in the city (daily)
- Inverse battle in the house found on Route 18 (daily - and you get a berry as well)
- Lumiose city Restaurants - Le Nah, Le Yeah, Le Wow and Sushi High Roller in South Boulevard, Autumnal Avenue, Hibernal Avenue and Hibernal Avenue respectively where you can have double battles, triple battles and even triple rotation battles. Also unlocked after restoring power to the city.
Post-ending of game
- Your rival - Found in Kiloude City (daily)
- Lumiose City - Looker Office (if the timing is right, you can battle Emma again)
- Elite Four (any [number of] time)