If the flu vaccine is the exact same as last year, would it still offer protection this year?
If one flu strain wouldn't change between the years (meaning there wouldn't be any new mutations in it) and the same strain would circulate again in the following year, the "old" vaccination would protect people. This is the same as with other viruses which do not change, once you reached a protection (usually one basic immunization and one booster), this protects for a long time (see measles vaccination as an example here).

There are surprisingly little articles available on how long the immune response after a vaccination lasts. There is a difference between the livettenuated influenza vaccine (LAIV) and the inactivated influenza virus (IIV), with the former giving a less good protection (see reference 1).
Data for the LAIV suggests that there is protection for at least one year after the vaccination (see reference 2).

However, there is evidence that the immunity against the flu after an infection against this specific strain might last life-long, as a paper demonstrated on survivors of the 1918 flu epidemic (see reference 3).

Having the flu against being vaccinated against it is a complete different thing (higher exposure, probably more antigens), but the question is if the immune response against vaccinations may also for a longer time.
The problem with the flu vaccination are:


The strain circulating in a year is changing.
The stains are mutating, making them different from the virus used for vaccine production. Depending on how different these mutations make the circulating strain, it can render the vaccine more or less useless. This is something what happened this flu season.
The flu vaccine is prepared from different flu strains to cover a broader range. In a tri-valent vaccine these are typically 2 Influenza A and one Influenza B virus strains, in tetra-valent vaccines 2 A and 2 B strains. The composition is based on a prediction (from epidimiologic data) which strains are likely to circulate in the next season and is done early in the year for the following season (production takes time, etc.).


Because of these problems, researchers are working on a universal flu vaccine; which is based on virus parts which do not change and should protect against all virus strains. Then only the basic vaccination would be necessary as with other vaccinations.

With all these limitations, it is still advisable to get a flu vaccination, as it is a pretty severe infection.

When not challenged (i.e. upon infections) antibodies are cleared from the blood within a few weeks, but there are low amounts produced (which are enough for detection). The immunologic memory is maintained by memory B cells which can ramp up production of the highly specific antibodies in a very short time - this is the reason for the protection by immunizations.

References:


Modest Waning of Influenza Vaccine Efficacy and Antibody Titers
During the 2007-2008 Influenza Season.
Longevity of B-cell and T-cell responses after live attenuated
influenza vaccination in children.
Neutralizing antibodies derived from the B cells of 1918 influenza
pandemic survivors