Partly true, partly not. Influenza viruses mutate rapidly so the virus that causes this years flu is different enough from last years influenza virus that immunity to last years influenza virus won't protect you from this years influenza virus. That is one reason you need to a flu shot every year. So far at least the mutations to the COVID-19 virus have not produced changes to the virus that would make current vaccines ineffective. How long the immunity to Covid-19 from the vaccine will last has not determined because it is so new however Moderna expects their vaccine to provide immunity for at least 1 year. Time will tell if that is the case or not.