Do Phal leaves normally grow like this?
 

Don't they usually come up from the crown? I don't really know what's going on here.

Looks like it's growing a basal keiki. New plants can sometimes form at the base of the existing plant. You'll probably end up with 2 plants in your pot

Lecent

What do I do now? The plant is pretty healthy overall and happily blooming

Just leave it alone and enjoy twice the plant.

Do I ever have to separate them?

Keep an eye on it, I had a P. Mini Mark 'Holm' that tried to grow 2 leaves at the same time, and they popped out of the side of the plant. I thought it was a basal keiki, and then it rips right up into the crown of the plant and it's two leaves. They did not grow right, and are small, but it's starting a new leaf from it's new crown. I could get a pic if you are interested. I have also seen spikes get stuck and bent, requiring a little help. Could you get a clearer pic of yours? Also you can separate or not, according to your preference.

I wouldnt separate, if its a new growth it will just add to the show when it blooms, you get two spikes instead of one, maybe even more if you plants are happy.

More questions...

1. Do I need to fertilize the plant more or take care of it differently?
2. When I repot, do I have to do anything special?

Mirranova, I have attached more pictures with different angles.

The pics are blurry. You shouldnt have any special requirements when repotting, if you want to keep the new growth connected just do what you do, if you want two plants from it just separate them, makesure they both have enough roots to separate them. Dont treat the plant any differently it happy this way.

Nah, it looks more like a basal keiki in these. Mine looked like something folded up popping out of the side. Pics are still blurry, must be from the resize. You can separate when the keiki has roots approx. 2 inches long. No extra special care, just do what you have been doing. If you keep them both together, just repot carefully so as not to injure the two plants. If you keep them together, it will make an incredible bloom display.