Interesting. So looks like without a facebook account, Tinder can't be installed? Think I'll have a look at POF instead!
No, you can actually just use your phone number. It used to be the case that you needed a Facebook account but not anymore. You're right that the text comes from Facebook and it's something called Account Kit by Facebook. However that's just something created by Facebook to allow people to sign in to apps and websites using their phone numbers (or in some cases email address). You can see more about it here:
External Link/Members OnlyQ: Does Account Kit share my Facebook information with the app or site?
A: No, Account Kit does not require a Facebook account and does not enable sharing of your information to the app or site you sign in to.
One might not believe/trust them but I have signed up to Tinder using a phone number Facebook couldn't possibly know about/link me to and it worked fine. As per the text you get sent, just use the code they send you.
You can of course just create a fake Facebook account (solely for the use of Tinder) as well. It's quick and you only need an email address to a sign up (something you can easily create as well). The only real advantage to that is that you can like things and if your potential match likes them too then it appears as mutual likes/interests. Of course though any influence of that actually has on their decision making will be tiny to non-existent.