Not sure if you are telling us or asking us. I have traveled with my dog many times. And if you really want to know - it varies by airlines.
Most airlines allow a limited about of small dogs/cats under a certain weight to travel in the cabin for a fee. Yes they are like carryon - and typically the requirement is that they are in an approved airline carrier in which has a water proof bottom (for obvious reasons) and that it can fit under the seat in front of you - you are not to take the animal out of the carrier. They have to have enough room where they can turn around in the carrier.
I am less familiar with the pet cargo requirements as I have never used them.
Many hotels allow dogs - again their rules vary as to amount of pets, size and almost all charge a fee for this.
If you are seriously interested in bringing your pet with you (especially dogs) this website gives lots of dog friendly places for traveling
https://www.bringfido.com/