i think you're just playing it too much. when we do something so much that it consumes our waking life, it can seep into our dreams. i would just take it as a fun thing, or cut back on the ff4 gameplay.
for a more psychoanalytical prospective, the door, it symbolizes opportunities before you, and has different meanings depending on how it opens. if you pull it open you desire more inner-exploration where as if you push it to open you wish to be more accessible to others. if you are inside the building you require introspective, but outside you desire freedom. since you're dreaming of a fantasy adventure game, i will theorize your dream is telling you that you desire something new and fun. you might be in the same old routine, and you feel trapped. you could shake it up a little by joining a club, walking home a different way, even stopping off at the park/cafe/library once a week could be enough of a change to make you feel better.
in regards to the portal question, no, you cannot teleport into an anime, videogame, book, or any fantasy world. you can look into astral projection, it might be what you're looking for. it's not ff4, but it's a higher plane where many 'fantasy creatures' dwell.