Bullshit. They are totally different things.
The vaccine enhances the immune system. It makes it better. Like tuning up an engine will make it run better. Like adding vitamins and minerals to certain foods make them more nutritious. Like giving you a brain will/might make you smarter.
I think the idea is that the stronger your immune system is better off you are going to able to ward off virus' and pathogens.
The people with the weakest immune systems seem to be the ones that have metabolic syndrome or pre diabetes, Unfortunately it is estimated that in the U.S. alone depending on who you're listening to that could be up to 60% of the population. Those groups being economically disadvantaged and racialized being even higher.
So in reality the message to manage your macronutrient and micro nutrient intake to achieve optimal health and thus develop your immune response to it's fullest should have been part of the message delivered from the beginning. That of course coupled with fresh air and exercise.
A doctor once said to me that if his patients were willing to make dietary and exercise changes he probably would hardly have to write a prescription. But sadly the general populace won't.