HECK NO! That is definitely NOT how the seat should support your weight! No wonder you are in pain.

A seat needs to have the proper width to support the two bony bumps on the inside of your glutes (one bump on each side). Those two bony bumps are your "sit bones," aka ischial tuberosities of your pelvic bone, where your weight can be supported without causing discomfort.

I would suggest you adjust the seat's angle and horizontal position (it can be slid forward or backward on its rails) to see if you can get the "wings" of the seat to support your "sit bones." If you can't find a comfortable position, then you most likely need a different saddle, one with a proper width for your butt.

I can sympathize. I had gone through five different saddles before I found a comfortable one (Serfas e-Gel for my particular butt, it fits me so nice I've done centuries on it, without need for padded shorts). Everyone has a slightly different butt, and you need to find one that is comfortable for you.