  Changes in planetary obliquity, or axial tilt, influence the climates on
Earth-like planets. In the solar system, the Earth's obliquity is stabilized
due to interactions with our moon and the resulting {small amplitude variations
($\sim$2.4\degree)} are beneficial for advanced life. Most Sun-like stars have
at least one stellar companion and the habitability of circumstellar exoplanets
is shaped by their stellar companion. We show that a stellar companion can
dramatically change whether {Earth-like obliquity stability is} possible
through planetary orbital precession relative to the binary orbit or resonant
pumping of the obliquity through spin-orbit interactions. We present a new
formalism for the planetary spin precession that accounts for orbital
misalignments between the planet and binary. Using numerical modeling in
$\alpha$ Centauri AB we show: a stark contrast between the planetary obliquity
variations depending on the host star, planetary neighbors limit the possible
spin states for {Earth-like obliquity stability}, and the presence of a moon
can destabilize the obliquity, defying our Earth-based expectations. An
Earth-like rotator orbiting the primary star would experience {small} obliquity
variations for 87\%, 74\%, or 54\% of Solar type binaries, depending on the
mass of the primary (0.8, 1.0, or 1.2 M$_\odot$, respectively). Thus,
Earth-like planets likely experience much larger obliquity variations, with
more extreme climates, unless they are in specific states, such as orbiting
nearly planar with the binary and rotating retrograde (backwards) like Venus.
