This page from the University of Oxford vaccine knowledge project says that flu vaccine contains traces of antibiotics and formaldehyde. They don't define "trace" but it usually means something well below the level that might have any effect, and probably so low that it can't be measured accurately.
This is likely to be something left over from the manufacturing process. At a guess, its because the virus used to make the vaccine needs to be cultured in live cells, and those cells are in a solution of antibiotics to prevent bacterial growth.
So the original claim is technically true, but presented in a misleading way. It implies that you will receive a significant dose of antibiotic along with the vaccine. This is false.