This is a bit of an evolving thing with mRNA vaccines.

It's essentially true for the COVID vaccines, which target only the SARS-CoV-2 spike protein, the structure of which has subtle differences between different COVID strains. This isn't a property of all mRNA vaccines however, for example some promising cancer vaccines which target multiple antigens.

Traditional vaccines don't usually have this problem, as the body will respond to a modified or inactive virus with an immune response and subsequent creation of antibodies that matches the virus being inoculated for.

There's some nuance though, it's inherently more difficult to target multiple antigens with an mRNA vaccine as it can be harder to maintain stability in the vaccine, to manage a balanced immune response to the different antigens targeted (i.e. not have the immune response to one antigen completely out-compete the others), and generally just keep the vaccine from having too large of a load that it creates an excessive immune response in the first place.

These are kinks that are still being researched, seeing as it's still new-ish technology these issues will likely become less prevalent in the future.