I have no idea where you read this, but it's not right. The immune system doesn't know anything about the brand of a vaccine. It remembers proteins and how to make antibodies to bind to proteins. Pretty much all vaccines are based on the exact RNA sequence of the spike protein from the first published COVID sequence. Immune systems can and do adapt to mutations in the spike protein by generating slightly different antibodies. It's how we fight colds and the flu, since these are constantly changing.

The main difference between the vaccine brands is the delivery mechanism. There were concerns in the beginning about mixing and matching; those were concerns about dosage, which was carefully determined in trials. Clearly a totally different mechanism screws with dosage. But nowadays I don't see many concerns about mixing and matching, particularly if just the booster differs from the (e.g. two first doses).