Are climbing shoes meant to be uncomfortable?
Climbing shoes were not designed to be uncomfortable, but, given their general nature of being tight, snug, and not leaving any wiggle room for your toes or any space to move for your feet, they are uncomfortable.
Though climbing shoes are quite uncomfortable, you do get used to the general fit of climbing shoes after some time. Normally, people wear general sport shoes that are very padded, loose, and comfortable, and they get used to wearing these kinds of shoes their whole life. Squeezing your feet into shoes that are almost the complete opposite of what you have been used to your entire life, can and will be very uncomfortable.
As I mentioned, you do get used to how climbing shoes feel. If you find the right shoes for you that are the right size and that fit the shape of your feet, the shoes can in fact feel much more comfortable than you would expect.