What is the genre / style of this song and songs like it? Short Piano Notes on every beat in background

In it's broadest sense, the term for this kind of repetitive accompaniment is ostinato: a repeated bass figure over which a melody occurs. Ostinato, however, can refer to a whole repeated sequence of notes or chords; it need not be a single chord.

Stylistically, the songs you're asking about have their roots in traditional New Orleans jazz. For example, listen to the guitar and drums in this recording. It's often called "four on the floor", which is a reference to a kick drum (bass drum) being played on all four beats of a measure, or more generally to the rhythm section as a whole.

Doctor John often uses this style, or variations on it, as does Randy Newman.

But short, repeated chords on each beat can show up in other styles. For example, Bohemian Rhapsody, "I see a little silhouette of a man...", owes more to classical music than jazz.

As PiedPiper noted, you tend to hear a version of this style frequently in Beatles' songs, such as "Hey Jude", "Penny Lane" and "Lady Madonna".

A great non-Beatles example of this style with more of the uptempo mood you're looking for would be "5th Beatle" Billy Preston's "Nothing from Nothing". It's certainly a distinctive sound, although I'm hard pressed to give it a name. It's also here in the work of another Beatles-adjacent artist, Jeff Lynne's "Mr Blue Sky".

It's basically 60s/70s piano rock, with the piano used as a rhythm instrument. You might call it "chords piano" or "rhythm piano" but neither of those is much good to search on. Your sample song puts me a quite a bit in mind of 70s Broadway, but I can't find a good example except maybe "Hard Knock Life" from Annie.

A different rhythm, but a similar feel and mood is Supergrass's 90s retro slacker anthem, Alright.

One classic use of this style of piano is on "A Day in the Life" by the Beatles from the "Sgt. Pepper" album. The link is to the part where that piano riff starts, but it's worth listening from the beginning.