That is true of VACCINE-ACQUIRED immunity, because the vaccine contains killed, attenuated or protein fragments specific to certain strains. These are inert substances incapable of invading cells thus are not perceived as a serious threat. Your body might produce IgM antibody that shows current or recent infection, but does not produce the subsequent IgG antibody that indicates current or PAST infection.
Natural immunity is created against a live, replicating virus that represents a threat to the body - the antibody is to the entire live organism and because of the memory of the cells that produce them they remain years after the original infection.
There are 4 types of flu viruses - A,B,C,D. Only Types A & B cause seasonal epidemics. The H1N1 and H3N2 are sub-types of Flu A and are generally considered more severe than Type B. Antibodies naturally acquired are specific to the Type (A or B), not the individual sub-types. All strains listed as sub-types under A or B are vulnerable.
Arup Lab Influenza A & B Virus Antibody
There have been only 3 flu seasons since 2009 that have had vaccine effectiveness greater than 50% - 2009-10 with 56%, 2010-11 with 60% (the highest reached), and 2013-14 with 52%. The current season VE for Type A is 37% and for Type B is 50%. I'm thinking that's not very impressive. Especially when you consider they only give percentages for those vaccinated. There is no data for infection rate of the UN-VACCINATED. If those percentages are found to be similar, I would surmise that the vaccine is basically useless.
But by all means, continue to get your yearly flu shot without first getting tested for naturally acquired Flu A and B Antibody. Go ahead and get the Coronavirus vaccine that has been rushed through WITHOUT the requisite animal studies and double blinds, followed by extensive followup for side effects, without testing first to see if you've developed the immunity naturally.
Any doctor/medical organization would be sued for malpractice or insurance fraud if they gave a drug to a patient without testing whether it is even necessary. Would a doctor prescribe insulin to someone just because they had a family history of diabetes and not because of an elevated A1C level, or chemotherapy to a patient just because there is family history or the patient's fear that he/she might get cancer?
What Immunity to COVID-19 Really Means
https://www.freerepublic.com/focus/f-news/3833679/posts
Can You Get Coronavirus Twice? How Long Are You Immune After COVID-19?
https://www.forbes.com/sites/brucelee/2020/03/15/can-you-get-infected-by-coronavirus-twice-how-does-covid-19-immunity-work/
Questions raised over COVID-19 reinfection after Japanese woman develops illness again
https://www.japantimes.co.jp/news/2020/02/28/national/coronavirus-reinfection/
Korean woman reinfected with coronavirus after recovery
https://www.koreatimes.co.kr/www/nation/2020/02/119_285335.html