Good code, just more complicated than needs to be. Helper functions don't need to return the object, you can directly modify the object within the scope of the function itself:

function solve(object, stringArray) {
    // const brName = object['Browser Name'];

    let newObj = {
        'Browser Name': object['Browser Name'],
        'Open Tabs': [...object['Open Tabs']],
        'Recently Closed': [...object['Recently Closed']],
        'Browser Logs': [...object['Browser Logs']],
    };

    for (let command of stringArray) {
        let data = command.split(' ');
        let currentCommand = data[0];
        let site = data[1];

        if (currentCommand === 'Open') {
            openTab(object, site);
            // openTab(object, site, newObj);
            // browserLogs(object, command, newObj);
        } else if (currentCommand === 'Close') {
            close(object, site);
            // close(object, site, newObj);
            // browserLogs(object, command, newObj);
        } else if (command === 'Clear History and Cache') {
            deleteObject(object);
        }
    }

    console.log(object['Browser Name']);
    console.log(`Open Tabs: ${object['Open Tabs'].join(', ')}`);
    console.log(`Recently Closed: ${object['Recently Closed'].join(', ')}`);
    console.log(`Browser Logs: ${object['Browser Logs'].join(', ')}`);

    function openTab(obj, site) {
        // function openTab(obj, site, secondObj) {
        // if (Object.keys(obj).length === 0) {
        //     obj = { ...secondObj };
        // } else {
        //     obj['Open Tabs'].push(site);
        // }
        // return obj;

        obj['Open Tabs'].push(site);
        browserLogs(object, 'Open ' + site);
    }

    function browserLogs(obj, command) {
        // function browserLogs(obj, command, secondObj) {
        // if (obj && Object.keys(obj).length === 0 && obj.constructor === Object) {
        //     obj = { ...secondObj };
        // } else {
        //     obj['Browser Logs'].push(command)
        // }

        obj['Browser Logs'].push(command)

        // return obj
    }

    function close(obj, site, secondObj) {
        // secondObj = Object.assign({}, obj);

        // if (secondObj['Open Tabs'].includes(site)) {
        //     let index = secondObj['Open Tabs'].findIndex(x => x === site);
        //     let result = secondObj['Open Tabs'].splice(index, 1);
        //     secondObj['Recently Closed'].push(result[0]);

        //     return secondObj;
        // }

        if (obj['Open Tabs'].includes(site)) {
            let index = obj['Open Tabs'].findIndex(x => x === site);
            let result = obj['Open Tabs'].splice(index, 1);
            obj['Recently Closed'].push(result[0]);
            browserLogs(object, 'Close ' + site);
        }
    }

    function deleteObject(obj) {
        // secondObj = Object.assign({}, obj);
        // delete secondObj['Browser Name'];
        // delete secondObj['Open Tabs'];
        // delete secondObj['Recently Closed'];
        // delete secondObj['Browser Logs'];

        // const emptyObject = {};
        // obj = { ...emptyObject };
        // return obj;

        obj['Open Tabs'] = [];
        obj['Recently Closed'] = [];
        obj['Browser Logs'] = [];

        // return obj;
    }
}