The purpose of GCPs is to improve the absolute accuracy of a 3D model (i.e. to locate it on Earth). In general, the accuracy of the outputs depends on the accuracy of the inputs given to Pix4Dmapper (i.e. the ground sampling distance/GSD), the accuracy of the image geotags (coordinates of the initial images), the accuracy of the GCPs measured in the field and the visual quality of the images.
The absolute accuracy of the outputs should be expected to be 1-2 pixels (GSD) horizontally (X, Y coordinates) and 2-3 pixels (GSD) vertically (Z coordinate). For instance, for a project with a GSD of 2 cm, the horizontal accuracy expected is about 2-4 cm and the vertical one 4-6 cm.
Please note that the DSM and DTM are models generated using other algorithms (point cloud densification, spatial interpolators). This results in a final model that is additionally refined and smoothed, thus modifying the underlying elevation values (including the Z values of GCPs). So, my answer to your question is: no, the elevation values of a DSM/DTM will often not coincide with the Z values from GCPs.
Hopefully this was helpful.