No, that's not possible to do with PDF files. You can use some kind of batch process to change all the files, but they will all have to be edited each time you want to implement this change.
Thanks for the reply - I am a little confused as you said a batch process could change all the files but then went on to say they will all have to be edited each time.
Do you mind elaborating? If there is a batch process to change all the files, is this third party software or something Adobe provides?
To make it more black and white, is there any way possible, with Adobe or any other program, to achieve the goal of creating one master file that, when changed, will change across multiple PDF files it has been assigned to be associated with?
A batch sequence (or Action as it is called in later versions of Acrobat) is a feature of Acrobat Pro that can be used to process multiple files in the same way. However, this is not a fully automatic and silent process. It needs to be triggered manually and it works by opening each file, executing the defined steps on it, and then closing it and moving on to the next one.
So this process can do what you described (ie, change all of your files in a specific way), but not in the way you described it (ie, you change one file and all the others are automatically changed accordingly).