No, the equivalent to jail-breaking an iDevice on an Android platform is:
Settings --> Applications --> Unknown sources (Place a checkmark there)
Rooting an Android device is so much more than jail-breaking an iDevice. It allows you to modify the very OS that is running, not just allowing you to install apps that some company doesn't approve of.
Except for the minutia, there is no technical difference between Rooting and Jailbreaking. Both techniques are designed to give the user root access to their devices.
I Jailbroke my iPad and that gave me the ability to access the root file system and delete anything I want to or need to delete. I can install apps from many different sources. I can add themes, change icons and icon badges, run applications Apple will not allow, like Frash, iFile, and I can add "Desktop" widgets. I can put programs and widgets on my lock screen and change that altogether.
I can do anything I want to do to the device include screw it up.
I can use flash drives and MicroSD Cards and access any file type I want to access directly from the card or drive. Clearly, this is not something Apple wants or they would have made it possible. Especially knowing that accessing doc, docx, and text files is something that some business users might want or need to do.
Txt and doc files cannot be accessed from a flash drive or MicroSD card with an unbroken device because only by jail breaking can I use the iFile File Manager. That allows me to open files not allowed by Apple's Camera Connection Kit.
When I go to "Settings --> Applications --> Unknown sources all that does is allow me to install foreign apps. And by no means is that remotely considered a JB; it is a standard feature that means very little. For example, how does that allow me to remove unwanted factory installed apps?
No, please forgive me, but you are incorrect in your understanding of JB and Apple products. My iPad is like a whole other device because it actually works for me and that would not be possible if not jailbroken.
I am not sure what you mean by modify the OS. Can you clarify? Not sure you can modify a compiled program unless you recompile it after making changes. Again, clarify. I can modify the OS to some extent like changing internal text files or images in the folders. For example, Apple's standard icon files. As for changing the code, it is beyond me and I'll say it, it is not at all possible or easy.
Bob