No, two notes, of the same denomination ($1), of the same series (2013), from the same (FRB) Federal Reserve Bank B-New York, is Ever supposed to have the same Serial Number, Ever, Ever, Ever, Regardless of being printed at two different facilities.
There are only Two Printing Facilities for all US Notes, that service all 12 FRB's. The BEP, issues printing orders for notes, authorizing how many, for what FRB, and which serial numbers can be used, etc.
If it made no difference because they were printed at different facilities, then they could purposefully print duplicated serial numbers and only need a 5 or 6 digit serial number, not 8, saving tons of money on ink and printing equipment. And there wouldn't be any need for Star Notes to begin with, if all they had to do was print a replacement note with the same serial number, just at a different facility.
No matter how you slice it, the BEP made a huge error having duplicated serial numbers printed, technically committing counterfeiting.
If something un-planned happens in the production of coins and currency and it makes it out into circulation, it's an Error.
This is an Error, even though PMG
won't use the word error at this point, but said they may change their mind in the future.
currently trying to ignore the fact, in calling this error an error, is a whole different story.
Hope that helps.