Tri-comb and multi-comb techniques could enable many advanced measurement applications beyond the reach of traditional dual-comb schemes. However, the sophisticated and bulky control systems of the conventional schemes based on three comb lasers render them impractical for many potential applications. Like their dual-comb counterparts, tri-comb and multi-comb lasers are being investigated as attractive alternatives. In contrast to previous dual-comb lasers using only one multiplexing dimension of optical pulses, this work simultaneously leverages multiplexing methods in three physical dimensions, i.e. wavelength, polarization, and direction, to generate triple to quadruple asynchronous pulse trains in a bidirectional mode-locked fiber laser. Because of the unique cavity structure studied here, both wavelength-multiplexed and polarization-multiplexed dual-comb generation from a completely shared-cavity and wavelength/polarization-multiplexed multi-comb generation from a bidirectional partially shared-cavity are achieved. Good relative stability among the generated combs of the fiber laser is demonstrated, as well as proof-of-concept dual-comb spectroscopy measurements, which validates the mutual coherence between the combs. The analysis of the experimental results further reveals interesting performance comparisons between combs from different multiplexing schemes, thanks to the special laser design used here that allows a side-by-side dual-comb demonstrations from different combinations of outputs from the same laser. Our investigation could facilitate multi-comb generation based on one light source for field-deployable multi-comb applications.