The subscribe() implementation seems suboptimal in the case where there are many different topics:

    subscribe(topic, callback) {
        if (this.listeners[topic] == undefined) {
            // Not yet any listener for this topic
            this.listeners[topic] = [];

            window.addEventListener('storage', (e) => {
                const dataKey = topic + "_data";
                if (e.key === dataKey) {
                    const data =  JSON.parse(e.newValue)[0];
                    this.listeners[topic].forEach((v, k) => {
                        v(data);
                    });
                }
            }, false);
        }
        this.listeners[topic].push(callback)
    }

This installs a handler for every single topic, and every time a message is published, the handlers for all topics are called, even though at most one is interested in the change. A more efficient implementation would install a single handler, e.g. (untested):

    window.addEventListener('storage', (e) => {
        if (e.key.endsWith('_data')) {
            const topic = e.key.substring(0, e.key.length - 5);
            const data = JSON.parse(e.newValue)[0];
            this.listeners[topic]?.forEach(v => v(data));
       }
    }, false);