Well, logically speaking, things such as your external feelings cannot affect a game - a machine incapable of thought. In the anime, it may be different. However, the sad reality is, the Pokémon we use, are just collections of data with pretty pictures.
However, if you grow attached to a certain Pokémon, you will know it more, and where it's flaws are, rather than someone who just copy and pasted Smogon's movesets. This is also where originality kicks in. Everyone thought Tangrowth was rubbish, but I had crowds watching my Round 3 match at VGC2009!
I do have my competitive teams, however, I also have the teams using Pokémon I love. I still hold a lot of respect for the Blaziken I used on Ruby, the Torterra I used on my Japanese Pearl, and the Staraptor I used in said Pearl. It's a nostalgia thing, we've been through a lot. Staraptor has defeated EV trained Pokémon with no problem - because I know his strengths and weaknesses through experience.
Same with DORES my Emboar; he was my starter in White, and I know a lot about the species now. Although base stats put ATK and Sp.Atk close together, the ATK stat sky rockets, and it takes a lot of damage from the recoil.
HANA, my Doreida, is strong on her own, but I know that she needs a little push; which is why when I battled Cynthia, I sent her out first, as I knew Spiritomb would spam Double Team, and I spammed Butterfly Dance - and she managed to take down 4 of her Pokémon before going down.
In HeartGold, I was determined that my in-game team was going to take down Red. I noticed my Slowking would end up duking it out with Red's Blastoise, and Blastoise would win due to speed. I taught Slowking Trick Room, and he won. For some reason, I really felt that Slowking wanted to defeat Blastoise, which is really bizarre, as they are just bits of data, and I wanted Slowking to win - I'd had it since it was a Slowpoke in Azalea Town, I wanted to give it it's honour.
Another case of me wanting to defend my Pokémon's honour was WILFRED my Mamoswine that I used in VGC10. My older brother kept on insulting it, saying how it never seems to win. I used it against him, and Mamoswine swept his whole team - landing a critical hit on the Lugia he wouldn't stop bragging about.
Today, in White, I challenged the Battle Subway singles, I used the previously mentioned DORES, YIPS my Waruvial, and CLAMP by Sazandra. I defeated the Brain without having to use Sazandra, who is a beast of a Pokémon anyway.
It's really difficult to say in words, but you really bond with your Pokémon. However, I don't think it's the bond itself that gives them strength, it's the experience you gain through using them and developing the bond. Remember, you can learn a lot about Pokémon from Smogon's pages and Skarmbliss' pages, but you can't learn how to play Pokémon without experience.