How should a climbing shoe feel?
A climbing shoe normally feels tight, snug, and applies a certain amount of pressure to your feet. This is especially true with new climbing shoes, as they have not yet been broken in. If the shoes are synthetic, they may not have a noticeable break in period, and however tight and snug they are when you buy them, may be how they feel even after a few months in them.
The same climbing shoes in the same size may also feel different for different people, as people have different shaped feet. The same shoes that may be applying a lot of pressure on your big toe, may be causing someone else some pain in another area of their foot. Everyone's feet are shaped differently and have different characteristics, so not all shoes will fit everyone the same, but, you do get used to most of the pressure after a short while.
Once you get used to wearing climbing shoes, a lot of the slight pains, discomforts, and other pressure points start to fade, and your feet get used to them.