Radar sensing has emerged in recent years as a promising solution for unobtrusive and continuous in-home gait monitoring. This study evaluates whether a unified processing framework can be applied to radar-based spatiotemporal gait analysis independent of radar modality. The framework is validated using collocated impulse-radio ultra-wideband (IR-UWB) and frequency-modulated continuous-wave (FMCW) radars under identical processing settings, without modality-specific tuning, during repeated overground walking trials with 10 healthy participants. A modality-independent approach for automatic walking-segment identification is also introduced to ensure fair and reproducible modality performance assessment. Clinically relevant spatiotemporal gait parameters, including stride time, stride length, walking speed, swing time, and stance time, extracted from each modality were compared against gold-standard motion capture reference estimates. Across all parameters, both radar modalities achieved comparably high mean estimation accuracy in the range of 85-98%, with inter-modality differences remaining below 4.1%, resulting in highly overlapping accuracy distributions. Correlation and Bland-Altman analyses revealed minimal bias, comparable limits of agreement, and strong agreement with reference estimates, while intraclass correlation analysis demonstrated high consistency between radar modalities. These findings indicate that no practically meaningful performance differences arise from radar modality when using a shared processing framework, supporting the feasibility of radar-agnostic gait analysis systems.