Dual-comb spectroscopy has emerged as an attractive spectroscopic tool for high-speed, high-resolution, and high-sensitivity broadband spectroscopy. It exhibits certain advantages when compared to the conventional Fourier-transform spectroscopy. However, the high cost of the conventional system, which is based on two mode-locked lasers and a complex servo system with a common single-frequency laser, limits the applicability of the dual-comb spectroscopy system. In this study, we overcame this problem with a bidirectional dual-comb fiber laser that generates two high-coherence ultra-broadband frequency combs with slightly different repetition rates (f rep). The two direct outputs from the single-laser cavity displayed broad spectra of > 50 nm; moreover, an excessively small difference in the repetition rate (< 1.5 Hz) was achieved with high relative stability, owing to passive common-mode noise cancellation. With this slight difference in the repetition rate, the applicable optical spectral bandwidth in dual-comb spectroscopy could attain ~479 THz (~3,888 nm). In addition, we successfully generated high-coherence ultra-broadband frequency combs via nonlinear spectral broadening and detected high signal-to-noise-ratio carrier-envelope offset frequency (f CEO) beat signals using the self-referencing technique. We also demonstrated the high relative stability between the two f CEO beat signals and tunability. To our knowledge, this is the first demonstration of f CEO detection and frequency measurement using a self-referencing technique for a dual-comb fiber laser. The developed high-coherence ultra-broadband dual-comb fiber laser with capability of f CEO detection is likely to be a highly effective tool in practical, high-sensitivity, ultra-broadband applications.