I believe Kirk's is a coconut soap and Dr. Bronner's is a vegetable blend soap, so they are not the same product other than that they are both liquid soaps. I wouldn't even call either one of them Castile, as I'm old school and believe the term Castile should be reserved for 100% olive oil soaps. So, depending on what kind of recipe you need liquid Castile for, you may need the 100% olive oil Castile or it might not matter what type of liquid soap you use.
From their ingredients lists, they are very different from each other. Dr. Bronner's ingredient list looks like real soap- the kind of liquid soap that myself and others here on the forum would make with oils and potassium hydroxide, etc..., while Kirk's is detergent-based with foaming agents.
Edited to add: Muskette is correct- neither is a true Castile, which is made with 100% olive oil.
If you're not able or willing to make liquid soap from scratch like Susie, Irish Lass, and many other soapers here, then you'll want to use the commercial "castile" product that your skin likes best, if you're making something to use on your body.
I have watched quite a few tutorials on making liquid Castille soap and iill admit, I've yet to make the jump into cold process soap. I wish I could back track and start there but I didn't. I've made a lot of products and that isn't one of them, maybe I'll try and find a good cold process first timer tutorial.