That is incorrect. You must talk to the trainer after the battle and they will ask if you want to register them.
Also, if you have Emerald, the easiest way to increase a Poke's Happiness rating is to feed them the stat-reducing berries (#21-#26). These berries peform two GREAT functions. The first is every time you feed one, it decreases the EVs of that stat by 10. This is so invaluable it isn't even funny. Now you can remove the crappy EV spreads that you have hurt your good Pokes with by playing through the storyline and can then focus on spot training and using EV training to your advantage. The second bonus is that every time you feed a Poke one of those berries, its Happiness level increases. Now, I have yet to calculate how many points each berry will increase (I is simple enough, just haven't bothered), but just remember that EVERY time you trade a Poke, its Happiness is reduced to 70. Yes, my friends, if your Snorlax with 220 Happiness is traded to another game, it drops all the way back to 70 and the fun begins again. Keep that in mind.