示意

如下图中的狐狸，鼠标左键按住可拖动，鼠标右键按住可调整大小，鼠标左键点击可随机播放一个动作。

思路

主要是2点

1. 让程序背景透明
   参考文章：Unity 制作萌系live2d桌宠：屏幕自适应+交互
   将TablePetBackSample.cs挂在一个GameObject上。
   将MainCamera的ClearFlags改为SolidColor，Background的颜色改为透明：000000

using UnityEngine;
using System;
// 为了使用属性：DllImport
using System.Runtime.InteropServices;

public class TablePetBackSample : MonoBehaviour
{
    private int currentX;
    private int currentY;
    #region Win函数常量
    private struct MARGINS
    {
        public int cxLeftWidth;
        public int cxRightWidth;
        public int cyTopHeight;
        public int cyBottomHeight;
    }

    [DllImport("user32.dll")]
    static extern IntPtr FindWindow(string lpClassName, string lpWindowName);
    [DllImport("user32.dll")]
    static extern int SetWindowLong(IntPtr hWnd, int nIndex, int dwNewLong);

    [DllImport("user32.dll")]
    static extern int GetWindowLong(IntPtr hWnd, int nIndex);

    [DllImport("user32.dll")]
    static extern int SetWindowPos(IntPtr hWnd, int hWndInsertAfter, int X, int Y, int cx, int cy, int uFlags);

    [DllImport("Dwmapi.dll")]
    static extern uint DwmExtendFrameIntoClientArea(IntPtr hWnd, ref MARGINS margins);
    [DllImport("user32.dll")]
    private static extern int SetWindowLong(IntPtr hWnd, int nIndex, uint dwNewLong);
    private const int GWL_EXSTYLE = -20;
    private const int GWL_STYLE = -16;
    private const int WS_EX_LAYERED = 0x00080000;
    private const int WS_BORDER = 0x00800000;
    private const int WS_CAPTION = 0x00C00000;
    private const int SWP_SHOWWINDOW = 0x0040;
    private const int LWA_COLORKEY = 0x00000001;
    private const int LWA_ALPHA = 0x00000002;
    private const int WS_EX_TRANSPARENT = 0x20;
    #endregion
    private IntPtr hwnd;

    void Awake()
    {
        var productName = Application.productName;
#if UNITY_EDITOR
#else
        hwnd = FindWindow(null, productName);
        int intExTemp = GetWindowLong(hwnd, GWL_EXSTYLE);
        SetWindowLong(hwnd, GWL_EXSTYLE, intExTemp | WS_EX_TRANSPARENT | WS_EX_LAYERED);
        SetWindowLong(hwnd, GWL_STYLE, GetWindowLong(hwnd, GWL_STYLE) & ~WS_BORDER & ~WS_CAPTION);
        currentX = 0;
        currentY = 0;
        SetWindowPos(hwnd, -1, currentX, currentY, Screen.currentResolution.width, Screen.currentResolution.height, SWP_SHOWWINDOW);
        var margins = new MARGINS() { cxLeftWidth = -1 };
        DwmExtendFrameIntoClientArea(hwnd, ref margins);     
#endif
    }
}

2. 关注鼠标的位置和点击
   将MyMouse.cs挂在一个GameObject上。

using System;
using System.Collections;
using System.Collections.Generic;
using System.Runtime.InteropServices;
using UnityEngine;


public class MyMouse : MonoBehaviour
{

    [DllImport("user32.dll")]
    public static extern short GetAsyncKeyState(int vKey);
    private const int VK_LBUTTON = 0x01; //鼠标左键
    private const int VK_RBUTTON = 0x02; //鼠标右键
    private const int VK_MBUTTON = 0x04; //鼠标中键

    private bool _isLeftDown;
    private bool _isRightDown;
    private bool _isMiddleDown;

    public event Action<MouseKey, Vector3> MouseKeyDownEvent;
    public event Action<MouseKey, Vector3> MouseKeyUpEvent;
    public event Action<MouseKey, Vector3> MouseDragEvent;
    public event Action<MouseKey> MouseKeyClickEvent;

    public Vector3 MousePos{ get; private set; }

    private bool _hasDragged;
    private Vector3 _leftDownPos;
    private Vector3 _rightDownPos;
    private Vector3 _middleDownPos;

    private void Start()
    {
        Init();
    }

    private void Update() 
    {
        // 按下左键
        if (GetAsyncKeyState(VK_LBUTTON) != 0)
        {
            if (!_isLeftDown)
            {
                _isLeftDown = true;
                _leftDownPos = MouseKeyDown(MouseKey.Left);
            }
            else if (MousePos != Input.mousePosition)
            {
                MouseKeyDrag(MouseKey.Left);
                if (!_hasDragged)
                {
                    _hasDragged = true;
                }
            }
        }
        // 按下右键
        if (GetAsyncKeyState(VK_RBUTTON) != 0)
        {
            if (!_isRightDown)
            {
                _isRightDown = true;
                _rightDownPos = MouseKeyDown(MouseKey.Right);
            }
            else if(MousePos != Input.mousePosition)
            {
                MouseKeyDrag(MouseKey.Right);
                if (!_hasDragged)
                {
                    _hasDragged = true;
                }
            }
        }
        // 按下中键
        if (GetAsyncKeyState(VK_MBUTTON) != 0)
        {
            if (!_isMiddleDown)
            {
                _isMiddleDown = true;
                _middleDownPos = MouseKeyDown(MouseKey.Middle);
            }
            else if(MousePos != Input.mousePosition)
            {
                MouseKeyDrag(MouseKey.Middle);
                if (!_hasDragged)
                {
                    _hasDragged = true;
                }
            }
        }
        // 抬起左键
        if (GetAsyncKeyState(VK_LBUTTON) == 0 && _isLeftDown)
        {
            _isLeftDown = false;
            MouseKeyUp(MouseKey.Left);

            if (!_hasDragged && _leftDownPos == MousePos)
            {
                MouseKeyClick(MouseKey.Left);
            }

            _hasDragged = false;
        }
        // 抬起右键
        if (GetAsyncKeyState(VK_RBUTTON) == 0 && _isRightDown)
        {
            _isRightDown = false;
            MouseKeyUp(MouseKey.Right);

            if (!_hasDragged && _rightDownPos == MousePos)
            {
                MouseKeyClick(MouseKey.Right);
            }

            _hasDragged = false;
        }
        // 抬起中键
        if (GetAsyncKeyState(VK_MBUTTON) == 0 && _isMiddleDown)
        {
            _isMiddleDown = false;
            MouseKeyUp(MouseKey.Middle);

            if (!_hasDragged && _middleDownPos == MousePos)
            {
                MouseKeyClick(MouseKey.Middle);
            }

            _hasDragged = false;
        }
    }

    private void OnDestroy()
    {
        Destroy();
    }
    
    public void Init()
    {
        _isLeftDown = false;
        _isRightDown = false;
        _isMiddleDown = false;
        _hasDragged = false;
    }
    
    public void Destroy()
    {
        
    }

    private Vector3 MouseKeyDown(MouseKey mouseKey)
    {
        RefreshMousePos();
        MouseKeyDownEvent?.Invoke(mouseKey, MousePos);
        // Log.Instance.ShowLog(string.Format("按下：" + mouseKey + "  "+MousePos));

        return MousePos;
    }
    private Vector3 MouseKeyUp(MouseKey mouseKey)
    {
        RefreshMousePos();
        MouseKeyUpEvent?.Invoke(mouseKey, MousePos);
        // Log.Instance.ShowLog(string.Format("抬起：" + mouseKey + "  " + MousePos));

        return MousePos;
    }

    private Vector3 MouseKeyDrag(MouseKey mouseKey)
    {
        RefreshMousePos();
        MouseDragEvent?.Invoke(mouseKey, MousePos);
        // Log.Instance.ShowLog(string.Format("拖动：" + mouseKey + "  " + MousePos));

        return MousePos;
    }

    private void MouseKeyClick(MouseKey mouseKey)
    {
        MouseKeyClickEvent?.Invoke(mouseKey);
        // Log.Instance.ShowLog(string.Format("点击：" + mouseKey + "  " + MousePos));
    }

    private Vector3 RefreshMousePos()
    {
        MousePos = Input.mousePosition;
        return MousePos;
    }

    public Vector3 MousePosToWorldPos(Vector3 mousePos)
    {
        var screenInWorldPos = Camera.main.WorldToScreenPoint(mousePos);
        var refPos = new Vector3(mousePos.x, mousePos.y, screenInWorldPos.z);
        var pos = Camera.main.ScreenToWorldPoint(refPos);
        return pos;
    }
}

public enum MouseKey
{ 
    None,
    Left,
    Right,
    Middle
}

如何确认是否点击到桌宠？
关注鼠标Down事件，发生时，获得当前鼠标位置mousePos

 var ray = Camera.main.ScreenPointToRay(mousePos);
 if (Physics.Raycast(ray, out RaycastHit hit))
 {
    // 判断hit的是否是桌宠。
 }