It is not normally necessary or recommended to have multiple flu vaccinations in the same flu season but if you do, it does not cause harm.
Flu vaccines protect against a number of flu strains; either 3-strains, trivalent flu jabs or 4-strains, quadrivalent flu jabs. In addition, they can also be “high-dose” or “adjuvanted” – flu vaccines approved for people 65 years and older to produce a greater immune response for more protection.
It is therefore possible that an older person or those with reduced immunity could have a high-dose flu vaccine followed by a standard dose of flu vaccine to enhance its effectiveness. A second dose, whether intentional or inadvertent, would not be expected to cause harm.