The DNA plasmids weren't picked up in the manufacturing process, they were inserted intentionally.
If they were picked up in the manufacturing process, they would show up in other medical products manufactured with the same equipment.
The DNA plasmids were purchased with the kanamycin insert, then updated with the SV40 promoter and spike protein sequence. The plasmid transfects an E.coli that replicates the plasmid and translates mRNA off the plasmid. Kanamycin is added to kill all the unsuccessfully transfected E. coli. A fresh batch of 100% transfected E. coli is grown. Part of the plasmid include a "poly A" tail on the mRNA. A magnetic reagent is added along with detergent to destroy the E. coli cell wall. The reagent has a poly-U structure to adhere to the poly-A. A magnet grabs all of the attached mRNA and the rest is supposed to be flushed away. A different buffer is added to release the mRNA with poly-A from the poly-U reagent. The reagent stays with the magnet for re-use. The mRNA is collected. At this stage a DNAase is supposed to be added to wipe out any remaining plasmid DNA. It was done incorrectly and lots of the plasmid DNA remained. The DNA and mRNA is then subjected to a multi step process to encapsulate in a lipid nano particle. The process wrapped both the intended mRNA and the DNA plasmid precursor. The LNP wrapping provided the dsDNA plasmid with access to transfect other cells in the target host.
If you have interest in finer details from people who did the actual lab work, see Kevin McKernan for plasmid details and Christie Laura Grace for the lipid encapsulation details. Both are available on X/Twitter.