Is it possible to play an electric guitar like a classical one?
Yes, and I regularly do so when warming up. I've also been known to play my Les Paul with my fingers at Big Band gigs. Mark Knopfler always used his fingers, and that riff from Smoke On The Water was played with the fingers (according to Total Guitar magazine).
The only real problem is that the electric neck is slimmer than the classical neck, so you don't have the space to isolate individual strings in the same way. Obviously, you don't get the top quality sound that you get from a proper classical guitar, and vibrato on an electric means bending strings (raising the pitch) rather than pushing (dropping the pitch) them but the notes are all in the same place.
I would recommend using the same playing position - shorten your strap to bring your electric up to the Hank Marvin "Geek Rock" position - as it keeps your elbow, wrist and fingers in more or less a straight line. Slinging your guitar low is ok when you're miming for the video but it puts an obscene amount of bad strain on your hand..
Why not just play it with your fingers as they are? Nobody uses finger picks in the classical world
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Sort of. You can certainly play an electric guitar fingerstyle... but you won't be able to play it exactly like a classical guitar.
There are several differences between the instruments. Alan has already touched on some, but a couple of other pretty big ones that matter are the action (electric guitars typically have a very low action) and the string spacing (how far apart the strings are at the bridge).
These two differences combine to make apoyando - the "rest stroke" of classical guitar - virtually impossible. You don't have enough room between the strings to use the same attack angle, and the low action means if you pull it off, you might be causing buzz against the frets and/or pickup poles.
I'm not sure if that's what you meant, though. When many guitarists talk about "classical" guitar style, they really mean fingerstyle. I consider "classical" a subset of fingerstyle, and the difference is that apoyando: classical guitarists all use it all the time to bring out certain notes, other fingerstyle approaches rarely do.
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Alan was pretty spot on. It's entirely possible to play classical guitar on an electric guitar, but it does feel a bit different. Though the differences are not all bad, in fact it can be a lot easier to play classical pieces on electric guitar since the nylon strings on classical guitars can sometimes be a bit harder to press down on. I find it's easier to play those really wide stretching positions on electric guitar.
The two downsides of playing classical guitar pieces on electric guitar are:
1. The tone is very different from nylon stringed classical guitars (but maybe you'd like this?)
2. You can potentially pick up bad habits which might make it awkward for you if you ever tried transitioning to an actual classical guitar.
All good advice above. I play classical (Grade 8 ABRSM) and also play a strat in a function band. I find it hard to do impromptu performances of any classical piece on my electric - but it can definitely be done. It's just a strange feeling because the action is so much lower and the neck narrower. With practice I'm sure you'd get used to this though. One further thing you might want to do is string your electric with higher gauge strings than you normally would so that there is a little more resistance and tension in the strings (and a fuller tone).
I take one small exception to Alan's comment, "The only real problem is that the electric neck is slimmer than the classical neck." This is mostly true, but there are some electric guitars that have pretty wide necks. I have a Gibson ES-225 and a Gibson ES-335, both hollow-body electrics, and their fingerboards are quite wide.
If by electric you mean the Stratocaster/Telecaster type of guitar, then I agree. I find the necks on those considerably narrower.
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Its possible alright. If you can do it, then its possible. It depends on what your aim with it is.
Trying new things out that you don't normally do is how you can develop your style of playing, and learn to express yourself in new variety of ways.
Take it or leave it, the shadow is always there. Holding your hand. With a glass of ice cold lemonade in the other. Listening to reggae music. What more do you want?