Is it possible to find objects that intersects arbitary 3d plane?

You could make your own meshcollider to find the objects in the plane like this (just example):

Add new GameObject then this component to it (notice that component requires MeshCollider in the same GameObject).

 [RequireComponent(typeof(MeshCollider))]
 public class CustomPlaneCollider : MonoBehaviour
 {    
     MeshCollider meshCollider;
     Mesh mesh;
 
     public void Start()
     {
         this.mesh = new Mesh();
         this.meshCollider = this.gameObject.GetComponent<MeshCollider>();
         this.meshCollider.sharedMesh = mesh;
         //just an initial plane.. set any values you like
         this.SetCollisionPlane(new Vector3(0, 0, 0), new Vector3(0, 0, 3.1f), new Vector3(4.3f, 0, 0), new Vector3(8.3f, 0, 5.3f));
         //triangles definition. This always stays same..
         this.mesh.triangles = new int[] { 0, 1, 2, 2, 1, 3 };
         this.ResetCollider();
     }
 
     //Set any coordinates for the vertices as you like. Call in Update or from ext components. Remember the plane is made of two triangles
     public void SetCollisionPlane(Vector3 p1, Vector3 p2, Vector3 p3, Vector3 p4)
     {
         this.mesh.vertices = new Vector3[] { p1, p2, p3, p4 };
         this.ResetCollider();    
     }
 
     private void ResetCollider()
     {
         this.meshCollider.isTrigger = false;
         this.meshCollider.convex = false;
     }
 
     //Testing...
     public void OnTriggerEnter(Collider other)
     {
         Debug.Log("Trigger enter: " + other.name);
     }
 }

You can call the SetCollisionPlane method in Update or from external code to set new position to plane.

Notice1. The plane is made of two triangles so it can bend depending how you set the coordinates.

Notice2. This example requires that your objects have collider set in Trigger mode & has Rigidbody.

Notice3. The GameObject you are using as host to this meshcollider sets the reference transform to the mesh also. By moving the GameObject the mesh and collider are moving also. Same goes for rotation and scaling. If you are moving the meshcollider that way then attach also Rigidbody to it (kinematic).

Hope this helps.