Brazilian bossa nova musicians were very much aware of, and influenced by, American jazz of the 1950s and "Golden Age" standards of the '20s to '50s. It's hard to miss the resemblance of "Batida's" bridge to "Satin Doll" - not only the chords, but also the starting note of each 4-bar phrase, and the fact that the second phrase is a repeated sequence of the first. Here's the bridge; the A section is in the key of G.
The chord pattern here (II V I to the key of the IV, then II V I up a step, leading to the V of the original key) is common enough in standards to have been dubbed the "Montgomery-Ward bridge." Nowadays we might have called it the "Walmart bridge."
The harmonic progression in the A section of "Batida" is pretty much the same as "Do Nothing 'til You Hear From Me," or "The Nearness of You."
Last week I brought in a chart of "Batida Diferente" for my Saturday adult combo class to play, and a couple of the band members noticed some other song resemblances. A stretch maybe, but the first phrase of the melody in "Batida's" A section isn't too far from this one:
"Be My Love" was composed in 1950 for Mario Lanza by Nicholas Brodzsky, lyrics by Sammy Cahn. It sold over 2 million copies. Ferreira and Einhorn certainly would have known the tune, but who knows?
The melody of the A section of "Batida" also resembles the original "Star Trek" theme (composed by Alexander Courage), but because the TV series didn't come out until 1966, it's disqualified as a "Batida" source. Maybe Courage listened to Mario Lanza (it's common knowledge that the harmony to "Star Trek" seems to have been lifted from Johnny Green's "Out of Nowhere").
Larry, our Saturday pianist, noticed that the bridge to "Batida" matches There! I've said It Again, a tune popularized by Vaughan Monroe in 1945, and also recorded by Jimmy Dorsey, Nat Cole, Bobby Vinton, and others. The chords are "Montgomery-Ward," but the melody also almost exactly matches in bar 1-2 and 4-5 of the bridge.