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.

yours look like having basal keiki. just keep an eye on it, when it's large enough to be separeted from mother plant, you may repot it.

but... while I was waiting mine to be mature enough, it started to bloom :) and later gave me some 3 other keikis, so, now I wait again :)

A basal keiki is a good thing! Especially if it comes when the mother plant is hale and hearty. When it matures, which it will much more quickly than a seedling plant, you'll have two flowering plants rather than one.

My personal preference is to keep basal keikis possted with their mother plant. I like the showiness of the extra blooming spikes. You take care of it like a single Phal, but maybe watching a bit more closely to make sure that all of the roots in the pot are being properly hydrated - two plants will obviously need more sustenance than one.

If you're going to split them, be very careful. Both plants must have a good, vigorous root system so that they can easily survive, and even thrive, as single plants. But, IMHO, it's more important to pay attention to HOW you divide the two. The keiki needs to have a strong, healthy stem/rhizome so that when you clip it off you won't cause damage to the plant. Further, when you do clip you need to make sure that the cut is done with sterile pruners and the cut area of the plant is sealed with some sort of fungal fighter. The easiest thing to use to seal the plant is powdered cinnamon. But again, this leads to a reason to keep the two plants together: it saves you the risk of cutting a Phal. Eventually, over a few years, the two will naturally grow apart and you can separate them far more easily.

Vina, those are beautiful!

Also, thanks for the advice everyone. Sounds like I'll leave it on there for awhile. I've also attached some pictures of the lucky parent (Phal. Sogo Tris)