This behavior, mainly visible in the phase graph, is due to the fact that both the myDAQ and NI ELVIS use multiplexed Analog Inputs, with the internal Analog to Digital Converter (ADC) being rapidly switched between two sources.
When you use the Bode Analyzer with a high-pass filter, one of the Analog Input channels is connected directly to the Analog Output channel of the device and the other Analog Input is connected to the capacitor, which impedance will be large at low frequencies and low a high frequencies.
When rapidly switching between a high impedance and a low impedance input, the result is distortion of the measured signal, with a greater differences between the signal amplitudes resulting in greater distortion. Since impedance and phase shift will decrease as frequency increases, this behavior is only seen at low frequencies.
This distortion does not seem to affect low-pass filters because at very low frequencies, the phase of a low-pass filter will be very close to zero, so this behavior will be masked.
Using a source impedance greater than 1 kΩ is not recommended when using any device with multiplexed Analog Inputs.