Hello,

This is a special case because you are trying to override an ES6 Class "getter" and unfortunately it will not work with the patch function.

I will propose you one solution (but  there  are  others) :

const patchMixin = require("web.patchMixin");
const PatchableMessage = patchMixin(components.Message);
const MessageList = require("mail/static/src/components/message_list/message_list.js");

PatchableMessage.patch(
    "owl_tutorial_extend_override_components/static/src/components/message/solution_3_patch_message.js",
    (T) => {    
        class MessagePatched extends T {      
            /**       
             * @override property       
              */      
            get avatar() {
                //  Code  your  override  here
            }
        }
        return MessagePatched;
    }
);
MessageList.components.Message  = PatchableMessage;

Basically, you force the patchMixin onto the original Component and you replace your patchable copy in the Component tree.
Here the Message Component is used in the MessageList component, so it is replaced here.

Be careful with that technique, if the original "Message" Component had other functions patched, all the patches will be lost !!

There are other ways to override "non-patchable" Components, via defineProperty or using the built-in "setup" function of 
Components. 

If you want to see other solutions, I recently wrote a tutorial on all the way to extend/override/monkey patch functions and took your question as an example, it may be helpful to you.

[Updated answer] Adding a patchInstance method

You can add a patchInstanceMethods function, inspired by the Odoo 15 version of the "patch" function. 

Create a new utils file inside your module: https://github.com/Coding-Dodo/owl_tutorial_extend_override_components/blob/main/static/src/js/utils.js

Now you can patch getters:

const {  patchInstanceMethods,} = require("your_module_name.utils");
patchInstanceMethods(components.Message.prototype, "messageFirstPatch", {  
    /**   
     * Get the avatar of the user. This function can be overriden   
     *   
     * @returns {string}   
     */  
    get avatar() {
        if (      
            this.message.author &&      
            this.message.author === this.env.messaging.partnerRoot    
        ) {      
            // example override
            return "https://avatars.githubusercontent.com/u/81346769?s=400&u=614004f5f4dace9b3cf743ee6aa3069bff6659a2&v=4";    
        } else if (this.message.author) {      
            return this.message.author.avatarUrl;    
        } else if (this.message.message_type === "email") {      
            return "/mail/static/src/img/email_icon.png";    
        }    
        return "/mail/static/src/img/smiley/avatar.jpg";  
    },
});
// SECOND PATCH
patchInstanceMethods(components.Message.prototype, "messageSecondPatch", {  
    /**   
    * Override of override   
    *   
    * @returns {string}   
    */  
    get avatar() {    
        let originAvatar = this._super(...arguments);    
        return originAvatar + "?overridenPatch=Yes";  
    },
});