Even though nowadays I know about how data for different Pokémon is made up, and how people can just easily generate Pokémon at will, I still have strong emotional attachments for Pokémon and see them as more than just data. I see them as living creatures and friends, and I've always been a sucker for the whole cheesy "connection/partnership between humans and Pokémon" moral that's so often referenced in the games and anime. Because of this, I often like to just simply visualize Pokémon I love, it just makes me happy in a dumb, cheesy way.
I know how the game's data is made up, and I know a bit about competitive battling, but I still develop attachments to Pokémon I catch and use in the main story regardless of nature, IVs, etc., and feel like I'm actually journeying alongside them on a grand adventure. Things like Amie/Refresh and the splash text in the beginning of battle + them looking back at you really make it a lot better for me, and I usually get a little upset when they faint.
Despite this, I do sometimes try to breed for a good nature/IVs/ability for a friend on my journey, but of course become attached to them afterward. Bringing up Pokémon from older generation games to the new ones is like bringing back old friends with memories of my past. Sometimes for important battles, I like for them to go a certain way (e.g. a certain Pokémon I want to shine delivering the finishing blow; it's pretty difficult to accept whenever it's difficult or impossible).
In battle simulators (Showdown) I can't really connect with the Pokémon emotionally because there is no time or memories in catching and raising them, and that they're quite obviously data that you can easily alter. In the games, I prefer to do as much as I can myself to acquire, raise, and spend time with my Pokémon. That's why if I ever get into breeding for competitive and/or shiny Pokémon, I'm able to do it all (or mostly) by myself. I'd like to be able to use Pokémon I love in the competitive stage, instead of having one bred or generated for me.
I truly think the games have gotten better over time in terms of letting people who love the emotional bond aspect of Pokémon actually do that. I hope it gets even better in the future.