The interruption of rehabilitation activities caused by the COVID-19 lockdown has significant health negative consequences for the population with physical disabilities. Thus, measuring the range of motion (ROM) using remotely taken photographs, which are then sent to specialists for formal assessment, has been recommended. Currently, low-cost Kinect motion capture sensors with a natural user interface are the most feasible implementations for upper limb motion analysis. An active range of motion (AROM) measuring system based on a Kinect v2 sensor for upper limb motion analysis using Fugl-Meyer Assessment (FMA) scoring is described in this paper. Two test groups of children, each having eighteen participants, were analyzed in the experimental stage, where upper limbs’ AROM and motor performance were assessed using FMA. Participants in the control group (mean age of 7.83 ± 2.54 years) had no cognitive impairment or upper limb musculoskeletal problems. The study test group comprised children aged 8.28 ± 2.32 years with spastic hemiparesis. A total of 30 samples of elbow flexion and 30 samples of shoulder abduction of both limbs for each participant were analyzed using the Kinect v2 sensor at 30 Hz. In both upper limbs, no significant differences (p < 0.05) in the measured angles and FMA assessments were observed between those obtained using the described Kinect v2-based system and those obtained directly using a universal goniometer. The measurement error achieved by the proposed system was less than ±1° compared to the specialist’s measurements. According to the obtained results, the developed measuring system is a good alternative and an effective tool for FMA assessment of AROM and motor performance of upper limbs, while avoiding direct contact in both healthy children and children with spastic hemiparesis.