Is there something like "did you forget to attach a file" in Apple Mail? In Thunderbird there is something like "are you sure you did not forget to attach a file" if you mentioned words such as "attached/in the attachment/ attach" and many more (could be set default on tb). 

Is there something similar for the Apple mail on OS X 10.9 Mavericks or for iPhone (iOS 6)?
For OS X, there is a beta version of an older free (donationware) plugin called Attachment Scanner Plugin for Mail.app that is compatible with OS X Lion and OS X Mountain Lion. It works similar to the one you have mentioned for Mozilla Thunderbird. Since OS X Mavericks was only recently released, you can test it to see if it works or contact the author.

On iOS though, there's nothing of this sort in the stock Mail.app. Due to application sandboxing, third party developers cannot develop plugins for any application (Apple's apps or otherwise). This can only be supported if Apple makes such a feature available in Mail.app. I do not know if third party mail apps have such a feature.