There's cross-reactivity, and they target both Influenza A and Influenza B strains (usually), so you'll get broader protection than strictly the strains most targeted by the vaccine. That said, you can ask the pharmacist for a copy of the FDA package insert, or google it yourself for the specific vaccine they're giving you. The package insert, under "DESCRIPTION" (Section 11), should tell you the precise strains that the antigens were developed from.
For example, google "FLUVIRIN fda package insert",and the one for the 2017/2018 flu season on page 8 you'll find "A/Singapore/GP1908/2015,IVR-180 (an A/Michigan/45/2015 (H1N1)pdm09- like virus; A/Hong Kong/4801/2014, NYMC X-263B (H3N2) (an A/Hong Kong/4801/2014- like virus); and B/Brisbane/60/2008, wild type (a B/Brisbane/60/2008-like virus)".
Google the same document for Fluzone quadrivalent, and you should find, for the 2020/2021 flu season, the strains "A/Guangdong-Maonan/SWL1536/2019 CNIC-1909 (H1N1), A/Hong Kong/2671/2019 IVR-208 (H3N2), B/Phuket/3073/2013 (B Yamagata lineage), and B/Washington/02/2019 (B Victoria lineage)" on pg 21.
In 2017, for example, the vaccine didn't work all that well, because of this exact problem. My dad, who lived in an eldercare facility, was confined to his room many days of the winter, which is something I sympathized with at the time, but sympathize a whole lot more with after this April's shelter-in-place order.
The deaths in the U.S. in 2017 were about 61,000, compared to the normal 40,000 or so.
Long story short: you don't know, but it's the best guess of those in charge of making that call.
I think if I was in your father's situation I would just go out and take my chances. At the age when I have to live in an eldercare facility I'll already be at a constant high risk of death from other causes. What's the point of avoiding the flu if it means I'm stuck in solitary confinement where a stroke might kill me tomorrow?
I think they meant when they go for their regularly scheduled -"usual" flu vaccine, and their question was meant to address what strains would be targeted in a given year.
For example, google "FLUVIRIN fda package insert",and the one for the 2017/2018 flu season on page 8 you'll find "A/Singapore/GP1908/2015,IVR-180 (an A/Michigan/45/2015 (H1N1)pdm09- like virus; A/Hong Kong/4801/2014, NYMC X-263B (H3N2) (an A/Hong Kong/4801/2014- like virus); and B/Brisbane/60/2008, wild type (a B/Brisbane/60/2008-like virus)".
Google the same document for Fluzone quadrivalent, and you should find, for the 2020/2021 flu season, the strains "A/Guangdong-Maonan/SWL1536/2019 CNIC-1909 (H1N1), A/Hong Kong/2671/2019 IVR-208 (H3N2), B/Phuket/3073/2013 (B Yamagata lineage), and B/Washington/02/2019 (B Victoria lineage)" on pg 21.