You can sign in on multiple devices with the same account. You must first have a registered account.
Checking if you have a registered account is easy. Open RecipeBox and tap on the Settings Tab. If you see at the very top a row that says "Register Account (Free)" you have an anonymous account. Tap on that row and quickly register an account. Once that is complete, you can sign into your RecipeBox account on multiple devices and in to the website.
If that Register Account row doesn't exist, you are ready to go, simply type your email address and your password into the other device and you will be signed in and all of your recipes will appear. If on the second device you are already signed in but can see the "Register Account (Free)" row, there will also be a "Log In" row directly below. Tap on "Log In", a pop up will appear where you can enter your email and password for RecipeBox. Once you have entered your email and password tap "Sign In" and all of your recipes will merge with your existing account.