It seems like "swearing" has been overloaded in your jurisdiction?
In my common law country, at least, when you swear something, it's an oath, and only an oath. We have a non-religious variant, no almighty God required, where you solemnly affirm to the same effect, and false information provided is perjury, punishable by up to 7 years imprisonment (if you perjured yourself in a judicial proceeding), unless you perjured yourself to get someone falsely convicted for a serious crime, then it's up to 14 years.
We also do statutory declarations, where you only declare it to true, but false information in one of those is still a crime, with a lesser penalty.
Oh, and we have a fun wee clause that maybe your jurisdiction doesn't? Where if you swear an oath, it can still be perjury even if it's not being sworn in a judicial proceeding. Only 5 years though.
So yeah, what does "swear" actually mean in the US? I don't get how you'd swear a DMCA takedown without your oath being taken. I figured if it was legit sworn, they'd do the usual "sworn this day X at BLA, witnessed/authorised/etc by Z"
Are they just trying to sound legal and scary, or is that format required in a takedown?
Not a lawyer, nor do I play one on TV, but I've been an occasional customer of the judiciary sufficiently to pick up the vagaries.