Background/Objectives: The aim of this study was to analyze the nonlinear dynamics of handgrip strength (HGS) in young adults, focusing on hand dominance, by employing the Poincaré plot method to assess short- and long-term variability utilizing dynamometry and video motion capture during sustained isometric contractions. Methods: A cross-sectional exploratory study was conducted on 30 healthy subjects (mean age 21.6 ± 1.3 years, 13 males and 17 females), measuring HGS for both the dominant hand (DH) and nondominant hand (NDH) using a Saehan hydraulic dynamometer during 25-s sustained isometric contractions. A GoPro HERO11 Black camera recorded the dynamometer’s needle movements, and the video data were analyzed using Kinovea software. Angular values were converted to force using a calibration-based formula, and the Poincaré plot computed variability indices (short-term variability—SD1, long-term variability—SD2, ratio SD1/SD2, and area of the fitting ellipse) for each hand in relation to HGS and angular velocity (AV). Data analysis included descriptive and inferential statistics. Results: We demonstrated a strong correlation between mechanical and video measurements (p ≤ 0.001), confirming the reliability of the video method. The findings highlight the importance of nonlinear analysis in understanding neuromuscular function and fatigue, revealing significant correlations among HGS, AV, Poincaré indices, and fatigue levels in both hands (p ≤ 0.001). Increased maximum HGS and AV correlated with higher nonlinear variability in force production. Conclusions: This study confirms the reliability of the proposed video-based HGS assessment and demonstrates the effectiveness of Poincaré plot analysis for capturing nonlinear variability in HGS.