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That sounds... a lot like the Our War Game section of the Digimon movie, but I doubt that would have been playing on TV every day! Maybe try the Cyberspace trope page and see if your show appears on it or any of its related trope pages?
Summer Wars?
A shot in the dark, but maybe Dennou Coil?
Call me "Heruru Meruru". http://blue-star-above-me.tumblr.com/C.L.Y.D.E.?
Bird instead of dog, but maybe Cyberchase?
I don't remember this cartoon that played almost everyday on my tv, I was so young but i remember that it had at least 1 boy and 1 girl and there was this "creature" (i guess it was a dog?) that didn't belong in the real world. It had like a virtual body, and it could enter inside the computer and then there was this huge "virtual highway" where all you could see emails being sent. I know that the boy was able to go inside the computer world with the creature and they had adventures there.
Edited by ColossalHelminthic