In the Embedded Service Deployment for Messaging for In-App Web, we can add the Logo in the Branding section.

I have used Asset File in Option 1. Check step 1 in the following link for your reference regarding Asset File.

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Option 1:

Option 2:

Using Salesforce Lightning Web Component, we can add logo to the Messaging for In-App and Web header. Use lightningSnapin__MessagingHeader as the target to set the Salesforce Lightning Web Component as the header in the Salesforce Embedded Service Deployment configuration for the Messaging for In-App and Web.

Sample Lightning Web Component for Header:

HTML:

<template>
    <section class={menuClass}>
        <template if:true={showMenuButton}>
            <button 
                class="headerButton menuButton" 
                onclick={onMenuButtonClick}>
                <lightning-icon 
                    icon-name="utility:threedots_vertical" 
                    variant="inverse" 
                    size="x-small">
                </lightning-icon>
            </button>
        </template>
        <section class="optionsMenu slds-dropdown slds-dropdown_left slds-nubbin_top-left">
            <ul class="slds-dropdown__list">
                <template 
                    for:each={chatbotOptionsMenu} 
                    for:item="option">
                    <li 
                        class="slds-dropdown__item" 
                        key={option.optionIdentifier}>
                        <button class="chatbotMenuOption"
                                value={option.optionIdentifier}
                                onclick={onMenuOptionClick}>
                            <span value={option.optionIdentifier}>
                                {option.title}
                            </span>
                        </button>
                    </li>
                </template>
                <li class="slds-dropdown__item">
                    <button class="chatbotMenuOption"
                            onclick={onEndConversationClick}>
                        <span class="slds-text-color_error">End Conversation</span>
                    </button>
                </li>
            </ul>
        </section>
    </section>
    <img 
        src="https://infallibletechiemiaworg--c.vf.force.com/resource/1730748259000/Logo" 
        alt="Logo">
    </img>
    <h2>Chat</h2>
    <button 
        class="headerButton minimizeButton" 
        onclick={onMinimizeButtonClick}>
        <lightning-icon 
            icon-name="utility:chevrondown" 
            variant="inverse" 
            size="x-small">
        </lightning-icon>
    </button>
    <template if:true={showCloseButton}>
        <button 
            class="headerButton closeButton" 
            onclick={onCloseButtonClick}>
            <lightning-icon 
                icon-name="utility:close" 
                variant="inverse" 
                size="x-small">
            </lightning-icon>
        </button>
    </template>
</template>

JavaScript:

import { LightningElement, api } from "lwc";
import { 
    dispatchMessagingEvent, 
    assignMessagingEventHandler, 
    MESSAGING_EVENT 
} from "lightningsnapin/eventStore";

const SLDS_MENU_SELECTOR = "slds-dropdown-trigger slds-dropdown-trigger_click";

export default class MessagingHeader extends LightningElement {

    @api configuration = {};
    @api conversationStatus;
    menuClass = SLDS_MENU_SELECTOR;
    chatbotOptionsMenu = [];

    get showMenuButton() {
        return this.conversationStatus === "OPEN";
    }

    _showMenu = false;
    get showMenu() {
        return this._showMenu;
    }

    set showMenu(shouldShow) {
        this.menuClass = SLDS_MENU_SELECTOR + (shouldShow ? " slds-is-open" : "");
        this._showMenu = shouldShow;
    }

    get showCloseButton() {
        return this.conversationStatus !== "OPEN";
    }


    onMenuButtonClick() {
        this.showMenu = !this.showMenu;
    }

    onMinimizeButtonClick() {
        dispatchMessagingEvent(MESSAGING_EVENT.MINIMIZE_BUTTON_CLICK, {});
    }

    onCloseButtonClick() {
        dispatchMessagingEvent(MESSAGING_EVENT.CLOSE_CONTAINER, {});
    }

    onMenuOptionClick(event) {

        event.preventDefault();
        dispatchMessagingEvent( MESSAGING_EVENT.MENU_ITEM_SELECTED, {
            selectedOption: this.chatbotOptionsMenu.find(
                option => option.optionIdentifier === event.target.getAttribute( "value" )
            )
        });
        this.showMenu = false;

    }

    onEndConversationClick() {

        dispatchMessagingEvent( MESSAGING_EVENT.CLOSE_CONVERSATION, {} );
        this.showMenu = false;

    }

    connectedCallback() {

        assignMessagingEventHandler(MESSAGING_EVENT.PARTICIPANT_JOINED, ( data ) => {

            console.log(
                'Participant joined'
            );
            if ( data.options && Array.isArray( data.options ) ) {
                data.options.forEach( ( participantOption ) => {
                    this.chatbotOptionsMenu.push( participantOption );
                } );
            }
        } );

        assignMessagingEventHandler( MESSAGING_EVENT.PARTICIPANT_LEFT, ( data ) => {
            console.log(
                'Participant left',
                JSON.stringify( data )
            );
            this.chatbotOptionsMenu = [];
        });

        assignMessagingEventHandler( MESSAGING_EVENT.UPDATE_HEADER_TEXT, ( data ) => {
            console.log(`Update header text: ${data.text}`);
        } );

        assignMessagingEventHandler(MESSAGING_EVENT.TOGGLE_BACK_BUTTON, ( data ) => {
            console.log(`Toggle back button visibility.`);

            this.showBackButton = data.showBackButton;
        } );

    }

}

CSS:

:host {
    width: 100%;
    max-width: 100%;
    min-height: 50px;
    display: flex;
    align-items: center;
    box-sizing: border-box;
    position: relative;
    padding: 0 14px;
    height: 50px;
    max-height: 50px;
    background-color: black;
}

img {
    margin-right: 4px;
    max-height: 40px;
    max-width: 40px;
}

h2 {
    margin: 0;
    text-align: initial;
    align-self: center;
    flex-grow: 1;
    overflow: hidden;
    text-overflow: ellipsis;
    white-space: nowrap;
    font-weight: lighter;
    font-size: inherit;
    color: white;
}

.headerButton {
    background: none;
    border: none;
    display: inline-flex;
    height: 32px;
    min-height: 32px;
    width: 32px;
    min-width: 32px;
    align-items: center;
    justify-content: center;
}

.headerButton:hover:before {
    content: " ";
    position: absolute;
    top: 9px;
    width: 32px;
    height: 32px;
    background-color: #fff;
    opacity: .2;
    border-radius: 4px;
    box-sizing: border-box;
    pointer-events: none;
}

.headerButton.menuButton:hover:before {
    top: 0;
}

.headerButton.closeButton:hover:before {
    right: 14px;
}

lightning-icon {
    fill: white;
}

.optionsMenu {
    width: 100vw;
    left: -0.9rem;
}

.optionsMenu > .slds-dropdown__list {
    max-height: 20rem;
    overflow-y: auto;
}

.slds-nubbin_top-left:before,
.slds-nubbin_top-left:after {
    left: 2rem;
    top: -0.5rem;
}

.slds-dropdown {
    max-width: 30rem;
}

.slds-dropdown__item > button {
    position: relative;
    z-index: 100;
    padding: 0.75rem 1rem;
    background-color: #ffffff;
    border: 0;
    display: flex;
    width: 100%;
}

.slds-dropdown__item > button:focus,
.slds-dropdown__item > button:hover {
    outline: 0;
    text-decoration: none;
    background-color: #e5e5e5;
}

.slds-dropdown__list > .slds-dropdown__item:not(:first-child) {
    border-top: #e2e2e2 1px solid;
}

.slds-dropdown__item .slds-truncate:not(.closeConversation) {
    color: #005290;
}

js-meta.xml:

<?xml version="1.0" encoding="UTF-8"?>
<LightningComponentBundle xmlns="http://soap.sforce.com/2006/04/metadata">
    <apiVersion>62.0</apiVersion>
    <isExposed>true</isExposed>
    <targets>
        <target>lightningSnapin__MessagingHeader</target>
    </targets>
</LightningComponentBundle>

Use the Lightning Web Component as the Header in the Custom UI Components of the Embedded Service Deployment.

Output: