iOS 4 and 5 were notorious for having this issue. They frequently ignored the ambient light sensor and were so stubborn, they refused to adjust the screen brightness to compensate. iOS 6 has corrected this issue.
The Apple MFI NDA requires that accessory developers not divulge any information about the device protocols, security chip, interface, etc.
However, device manufacturers are allowed to create libraries and SDKs that others can use to release apps that use their device.
One example is Redpark, for instance I have a few of their serial cables:
http://www.redpark.com/c2db9.html
You have access to as much of the device as they permit through their SDK, and you can release apps using that SDK that works with their device. You can search the app store for "redpark" and find many apps that use their various iOS cables.
So yes, it is possible - nothing in the Apple NDA prevents them from disclosing and providing their API to others. They simply cannot disclose Apple's protocol/API/interface to others.
Do note that Apple requires a company to submit device samples and use cases for MFI approval, and it appears they don't allow many developers to develop devices for which they plan to release SDKs. There have been efforts to create universal infrared remote accessories for instance that haven't been opened even though the developers originally intended to provide an SDK. One I discussed this with indicated that Apple wouldn't approve their device unless it was tied to one, and only one app - theirs. I understand in cases such as redpark the developers made the cables for specific customers and specific apps first, then started making generic cables for generic apps after initial approval.
So there still may be some reluctance on the part of Apple to allow developers to design open API devices.
While your question is about MFi accesories, a number of third party accessories have come out which are not MFi. Many use Wifi or the audio jack to communicate with their iOS apps, such as http://www.60beat.com/ . Several credit card readers plug into the audio jack. Not all of them give their SDK/API freely, but you can always ask.
Best Answer
So iPhone 5s use a Lightning cable rather than the old school dock connector.
Lightning cables actually have an authentication chip in them, hence the message.
There's a jailbreak tweak out there that may be worth checking out, but it's quite old and may not have been updated.
Your best bet would be to just get a new cable. Certified cables can be had for $8 on Amazon, and you don't run any risks of having dodgy cable damage your phone.
It's not just Apple protecting their profits, either. Certification actually means standards were followed.