How to get the process ID or name of the application that has updated the clipboard?

You can call GetClipboardOwner() to get the handle of the Window that last set or cleared the Clipboard (the operation that triggered the notification).

[...] In general, the clipboard owner is the window that last placed data in the Clipboard.
The EmptyClipboard function assigns Clipboard ownership.

There are special cases when a Process passes a null handle to OpenClipboard(): read the Remarks section of this function and the EmptyClipboard function.

Before calling EmptyClipboard, an application must open the Clipboard by using the OpenClipboard function. If the application specifies a NULL window handle when opening the clipboard, EmptyClipboard succeeds but sets the clipboard owner to NULL. Note that this causes SetClipboardData to fail.

► Here I'm using a NativeWindow derived class to setup a Clipboard listener. The Window that process the Clipboard update messages is created initializing a CreateParams object and passing this parameter to the NativeWindow.CreateHandle(CreateParams) method, to create an invisible Window.
Then override WndProc of the initialized NativeWindow, to receive WM_CLIPBOARDUPDATE notifications.

The AddClipboardFormatListener function is used to place the Window in the system Clipboard listeners chain.

► The ClipboardUpdateMonitor class generates an event when a Clipboard notification is received. The custom ClipboardChangedEventArgs object passed in the event contains the Handle of the Clipboard Owner, as returned by GetClipboardOwner(), the ThreadId and ProcessId returned by GetWindowThreadProcessId() and the Process name, identified by Process.GetProcessById().

You can setup a ClipboardUpdateMonitor object like this:
_{This class can also be initialized in Program.cs}

private ClipboardUpdateMonitor clipboardMonitor = null;
// [...]

clipboardMonitor = new ClipboardUpdateMonitor();
clipboardMonitor.ClipboardChangedNotify += this.ClipboardChanged;
// [...]

private void ClipboardChanged(object sender, ClipboardChangedEventArgs e)
{
    Console.WriteLine(e.ProcessId);
    Console.WriteLine(e.ProcessName);
    Console.WriteLine(e.ThreadId);
}

using System.Diagnostics;
using System.Runtime.InteropServices;
using System.Security.Permissions;
using System.Windows.Forms;

public sealed class ClipboardUpdateMonitor : IDisposable
{
    private bool isDisposed = false;
    private static ClipboardWindow window = null;
    public event EventHandler<ClipboardChangedEventArgs> ClipboardChangedNotify;

    public ClipboardUpdateMonitor()
    {
        window = new ClipboardWindow();
        if (!NativeMethods.AddClipboardFormatListener(window.Handle)) {
            throw new TypeInitializationException(nameof(ClipboardWindow), 
                new Exception("ClipboardFormatListener could not be initialized"));
        }
        window.ClipboardChanged += ClipboardChangedEvent;
    }

    private void ClipboardChangedEvent(object sender, ClipboardChangedEventArgs e) 
        => ClipboardChangedNotify?.Invoke(this, e);

    public void Dispose()
    {
        if (!isDisposed) {
            // Cannot allow to throw exceptions here: add more checks to verify that 
            // the NativeWindow still exists and its handle is a valid handle
            NativeMethods.RemoveClipboardFormatListener(window.Handle);
            window?.DestroyHandle();
            isDisposed = true;
        }
    }

    ~ClipboardUpdateMonitor() => Dispose();

    private class ClipboardWindow : NativeWindow
    {
        public event EventHandler<ClipboardChangedEventArgs> ClipboardChanged;
        public ClipboardWindow() {
            new SecurityPermission(SecurityPermissionFlag.UnmanagedCode).Demand();
            var cp = new CreateParams();

            cp.Caption = "ClipboardWindow";
            cp.Height = 100;
            cp.Width = 100;

            cp.Parent = IntPtr.Zero;
            cp.Style = NativeMethods.WS_CLIPCHILDREN;
            cp.ExStyle = NativeMethods.WS_EX_CONTROLPARENT | NativeMethods.WS_EX_TOOLWINDOW;
            this.CreateHandle(cp);
        }
        protected override void WndProc(ref Message m)
        {
            switch (m.Msg) {
                case NativeMethods.WM_CLIPBOARDUPDATE:
                    IntPtr owner = NativeMethods.GetClipboardOwner();
                    var threadId = NativeMethods.GetWindowThreadProcessId(owner, out uint processId);
                    string processName = string.Empty;
                    if (processId != 0) {
                        using (var proc = Process.GetProcessById((int)processId)) { 
                            processName = proc?.ProcessName;
                        }
                    }
                    ClipboardChanged?.Invoke(null, new ClipboardChangedEventArgs(processId, processName, threadId));
                    m.Result = IntPtr.Zero;
                    break;
                default:
                    base.WndProc(ref m);
                    break;
            }
        }
    }
}

Custom EventArgs object used to carry the information collected about the Clipboard Owner:

public class ClipboardChangedEventArgs : EventArgs
{
    public ClipboardChangedEventArgs(uint processId, string processName, uint threadId)
    {
        this.ProcessId = processId;
        this.ProcessName = processName;
        this.ThreadId = threadId;
    }
    public uint ProcessId { get; }
    public string ProcessName { get; }
    public uint ThreadId { get; }
}

NativeMethods class:

internal static class NativeMethods
{
    [DllImport("user32.dll", SetLastError = true)]
    [return: MarshalAs(UnmanagedType.Bool)]
    internal static extern bool AddClipboardFormatListener(IntPtr hwnd);

    [DllImport("user32.dll", SetLastError = true)]
    [return: MarshalAs(UnmanagedType.Bool)]
    internal static extern bool RemoveClipboardFormatListener(IntPtr hwnd);

    [DllImport("user32.dll")]
    internal static extern IntPtr GetClipboardOwner();

    [DllImport("user32.dll", SetLastError = true)]
    internal static extern uint GetWindowThreadProcessId(IntPtr hWnd, out uint lpdwProcessId);

    internal const int WM_CLIPBOARDUPDATE = 0x031D;

    internal const int WS_CLIPCHILDREN = 0x02000000;
    internal const int WS_EX_TOOLWINDOW = 0x00000080;
    internal const int WS_EX_CONTROLPARENT = 0x00010000;
}