It is better if you at least start out with a vague notion or idea, of what you would like to do next. However, I think it is best if you stick with the first one, as it may need some tweaking, and overall work. If you do two at the same time, you may drop both because of how they are going, or something like that.
Also some people start with characters and their personality before they move into conflicts, because they can see how their character might react to said conflict.
If this is your first fanfiction, I recommend you stick with it without starting a new one. After I started my first fanfiction, I decided I'd do another one as well. It didn't work so well; I ended up dropping both. Ehehe.
Yeah, because if you have too many to do, it will become rushed in a desperado to finish them all, and it wont end up so good. I'd finish your first one, then take a little time planning, thinking, and drafting a couple of paragraphs for a new story, before you get it right.
I expect a story, no-one, or at least hardly anyone has done before. I'd say that there should be many noticeable differences between other fanfics you made as well.
This is nigh on impossible. The most you can realistically hope for is someone who puts a new spin on an old idea. Of course, new things are possible, but the chances of them cropping up are very slim.
If you want to be pedantic and general, yes, every story is different. Anyone can do that. What very few people can do is make a story that is noticeably different. Actually coming up with something entirely new is incredibly difficult.
Particularly in fanfiction. Particularly in Pokémon Fanfiction.
Of course, since it is still theoretically possible, one could simply say Emp395 has high standards. C:
To answer the original question of the thread, though, I would recommend you don't try to jump into a second one. It's far easier to write one fic at a time, and it also gives you incentive to continue on with it (whereas if you have two, it's easier to just decide to drop one, rather than make it better.)