Camera.SetStereoProjectionMatrix not working as expected

Late answer, I guess you won't need the information anymore, but maybe it is helpful for someone else. I don't know if this is a bug, in any way it is weakly documented, but I am glad that Unity has this feature.

I use the non head-mounted stereo mode, but I am sure it is adoptable for the split-screen stereo mode. To set the projection matrix, I basically do the following:

 private void SetProjectionMatrix(Camera cam, Matrix4x4 projectionMatrix, Camera.StereoscopicEye? specifyBothEyeTarget = null)
 {
 #if UNITY_EDITOR // Editor only
     cam.projectionMatrix = projectionMatrix;
 #elif UNITY_STANDALONE // Build only
     string loadedDeviceName = null;
 #if UNITY_5 || UNITY_2017_1
     if(UnityEngine.VR.VRSettings.enabled == true)
     {
         loadedDeviceName = UnityEngine.VR.VRSettings.loadedDeviceName;
     }
 #else // #elif UNITY_2017_2_OR_NEWER
     if (UnityEngine.XR.XRSettings.enabled)
     {
         loadedDeviceName = UnityEngine.XR.XRSettings.loadedDeviceName;
     }
 #endif // End of version difference.
     if (loadedDeviceName.Equals("stereo")) // -vrmode stereo
     {
         if (cam.stereoTargetEye == StereoTargetEyeMask.Left)
         {
             cam.SetStereoProjectionMatrix(Camera.StereoscopicEye.Left, projectionMatrix);
         }
         else if (cam.stereoTargetEye == StereoTargetEyeMask.Right)
         {
             cam.SetStereoProjectionMatrix(Camera.StereoscopicEye.Right, projectionMatrix);
         }
         else // None or both:
         {
             if (specifyBothEyeTarget != null)
             {
                 // Setting left or right eye for "both-eye" camera, depending on the specifyBothEyeTarget.Value
                 cam.SetStereoProjectionMatrix(specifyBothEyeTarget.Value, projectionMatrix);
             }
         }
     }
     else if(string.IsNullOrEmpty(loadedDeviceName)) // -vrmode none or vr completely disabled
     {
         cam.projectionMatrix = projectionMatrix;
     }
 #endif // End of UNITY_STANDALONE only
 }

You can call this method dependently if you are using a single both-eye camera or two separate cameras. When calling it, for a both-eye cam, you need to call it twice (once for the left and once for the right side) and set the 3rd parameter (StereoscopicEye). If you call it for a left or right eye camera, just call it once and leave the 3rd parameter blank, the camera knows its target eye.
This also answers your other question: Yes, you need to set different projection matrices when using a camera with target eye both or when using Single Pass rendering. (Using separate cameras means Multi Pass rendering implicitly.)

An additional note: If you work / develop on a machine with a video card that does not support 120Hz stereo (e.g. Nvidia GTX, RTX, etc.), you can still use the "-vrmode stereo" command line argument. This will of course not really activate stereo rendering, but in this case, the projection matrix can be set with cam.SetStereoProjectionMatrix(), whereas, if the application is started with "-vrmode none", the projection matrix must be set with cam.projectionMatrix. The above written code takes care of these things.

All this is still true for Unity 2018.3.6 (latest version at the time of writing).

Additionally, you can quickly try out this demo script:

 using UnityEngine;

 [RequireComponent(typeof(Camera))]
 public class ProjectionMatrixTest : MonoBehaviour
 {
     private Camera _cam;

     private Matrix4x4 _projMat;
     private Matrix4x4 _projMat2;

     private void Awake()
     {
         _cam = GetComponent<Camera>();
     }

     private void Start()
     {
 #if UNITY_EDITOR
         _projMat = _cam.projectionMatrix;
 #else
         if (_cam.stereoTargetEye == StereoTargetEyeMask.Left)
         {
             _projMat = _cam.GetStereoProjectionMatrix(Camera.StereoscopicEye.Left);
         }
         else if(_cam.stereoTargetEye == StereoTargetEyeMask.Right)
         {
             _projMat = _cam.GetStereoProjectionMatrix(Camera.StereoscopicEye.Right);
         }
         else // Both or None
         {
             //_projMat = _cam.projectionMatrix;

             _projMat = _cam.GetStereoProjectionMatrix(Camera.StereoscopicEye.Left);
             _projMat2 = _cam.GetStereoProjectionMatrix(Camera.StereoscopicEye.Right);
         }

 #endif
     }

     private void Update()
     {
         Debug.Log("Updating: " + _cam.stereoTargetEye.ToString());

         _projMat.m01 += Mathf.Sin(Time.time * 1.2F) * 0.1F;
         _projMat.m10 += Mathf.Sin(Time.time * 1.5F) * 0.1F;
         _projMat.m21 += Mathf.Sin(Time.time * 0.8F) * 0.1F;

         _projMat2.m01 += Mathf.Sin(Time.time * 1.2F) * 0.1F;
         _projMat2.m10 += Mathf.Sin(Time.time * 1.5F) * 0.1F;
         _projMat2.m21 += Mathf.Sin(Time.time * 0.8F) * 0.1F;
         

 #if UNITY_EDITOR
         _cam.projectionMatrix = _projMat;
 #else
         if (_cam.stereoTargetEye == StereoTargetEyeMask.Left)
         {
             _cam.SetStereoProjectionMatrix(Camera.StereoscopicEye.Left, _projMat);
         }
         else if(_cam.stereoTargetEye == StereoTargetEyeMask.Right)
         {
             _cam.SetStereoProjectionMatrix(Camera.StereoscopicEye.Right, _projMat);
         }
         else // Both or None
         {
             //_cam.projectionMatrix = _projMat; // This does not work.
             
             _cam.SetStereoProjectionMatrix(Camera.StereoscopicEye.Left, _projMat);
             _cam.SetStereoProjectionMatrix(Camera.StereoscopicEye.Right, _projMat2);
         }

 #endif
     }
 }

Just add it to a camera (Single- or both-eye) and add a huge cube in front of the camera, so that you can see the effect better. The projection matrix gets messed up, but you can see, that the API works. If you add it to a both-eye camera and comment out one of the two projection matrix apply lines (SetStereoProjectionMatrix...) and create a build, you can see, that only one eye goes wild, but the other one stays normal.