Dual‐comb spectroscopy is a rapidly developing technique enabling ultraprecise broadband optical diagnostics of atoms and molecules. This powerful tool typically requires two phase‐locked femtosecond lasers, yet it has been shown that it can be realized without any stabilization if the combs are generated from a single laser cavity. Still, unavoidable intrinsic relative phase fluctuations always set a limit on the precision of any spectroscopic measurements, hitherto limiting the applicability of bulk dual‐comb lasers for mode‐resolved studies. Here, a versatile concept for low‐noise dual‐comb generation from a single‐cavity femtosecond solid‐state laser based on intrinsic polarization‐multiplexing inside an optically anisotropic gain crystal is demonstrated. Due to intracavity spatial separation of the orthogonally polarized beams, two sub‐100 fs pulse trains are simultaneously generated from a 1.05 µm Yb:CNGS (calcium niobium gallium silicate) oscillator with a repetition rate difference of 4.7 kHz. The laser exhibits the lowest relative noise ever demonstrated for a bulk dual‐comb source, supporting free‐running mode‐resolved spectroscopic measurements over a second. Moreover, the developed dual‐comb generation technique can be applied to any solid‐state laser exploiting a birefringent active crystal, paving the way toward a new class of highly coherent, single‐cavity, dual‐comb laser sources operating in various spectral regions.