Our study demonstrates second harmonic generation (SHG) in a high confinement LiNbO₃ rib waveguide through type-I birefringence phase matching of fundamental modes. The combination of micro-waveguide dispersion and material birefringence reveals unique SHG characteristics that complement the performance of standard LiNbO₃ on insulator (LNOI) components. Dual-wavelength phase matching in the NIR and MIR regions is revealed theoretically using a single waveguide cross-section. These two wavelengths can be positioned in the 1.1 μm-3.5 μm range or converge near 1.5 μm by adjusting the core waveguide size or through temperature tuning. A 25 °C temperature change enables a broad pump tunability band of 300 nm, ranging from 1350 nm to 1650 nm, with a conversion efficiency exceeding 40 %/W/cm$^2$ within a single waveguide. A temperature tuning range up to 900 nm is foreseen by tailoring the waveguide core size. In addition, a broadband response of 150 nm within the telecom window is demonstrated experimentally. The nonlinear waveguide, etched in a thin film membrane, is combined with titanium-indiffused waveguides to form a monolithic LiNbO₃ component. This configuration provides low coupling losses of 0.8 dB and single-mode operation. It paves the way for a new generation of versatile and cost-effective frequency conversion components with wide-band spectral responses suitable for optical communications, environmental sensing, and quantum information processing.