Dexterous robotic hands require high-speed multimodal sensing across many degrees of freedom, yet existing readout architectures often impose trade-offs between sensor count, wiring complexity, and sampling bandwidth. This paper presents a scalable analog sensor readout architecture based on a serial-in parallel-out (SIPO) shift-register principle. The proposed architecture supports versatile integration of heterogeneous analog-output sensors, scalable expansion using only three signal lines between sensor modules, and fast, configurable sampling. We validate the approach on a tendon-driven robotic hand integrating 16 joint sensor modules and one four-channel tactile sensor module, enabling acquisition of 20 sensor channels at a full-scan rate of 1 kHz, with stable operation up to 1.5 kHz. Joint sensor characterization showed a maximum slope absolute percentage error (APE) of 0.446% and sub-degree estimation error, indicating that the proposed readout system does not significantly degrade sensing performance. For tactile sensing, LSTM-based models achieved an RMSE of 0.125 N for force estimation and 93.4% accuracy for five-class contact-location classification, and were deployed for real-time inference at 1 kHz. System-level experiments showed that the joint sensors provide more accurate feedback than motor-based estimation during interaction, while the tactile sensor enables responsive force estimation in contact. The proposed architecture offers a practical path toward fully sensorized robotic hands for dexterous manipulation.