Mode-locked fiber lasers that can simultaneously generate two asynchronous ultrashort pulse trains could play an attractive role as the alternative light sources for low-complexity dual-comb metrology applications. Although a few multiplexing schemes to realize such lasers have been proposed and demonstrated, here we investigate the lasing characteristics of a passively mode-locked fiber laser with a finite amount of intracavity birefringence. By introducing a section of polarization-maintaining (PM) fiber into the otherwise-non-PM-single-mode cavity, dual asynchronous pulses with nearly orthogonal states of polarization are generated. With a repetition rate difference of hundreds of hertz, the pulses have well-overlapped spectra and show typical features of polarization-locked vector solitons. It is demonstrated that under an anomalous or net normal dispersion regime, either dual vector solitons or dual dissipative vector solitons can be generated, respectively. Such polarization-multiplexed single single-cavity dual-comb lasers could find further uses in various applications in need of simple dual-comb system solutions.