You can run content scripts on the Youtube Music website & app and parse the HTML to get the info you need. Here's a quick script with lazy code. You should obviously do the detection and error-handling better.

Manifest:

{
    "manifest_version": 3,
    "name": "Youtube Music Info",
    "version": "1.0",
    "description": "Gets information from YT Music.",
    "content_scripts": [
        {
            "matches": [ "https://music.youtube.com/*" ],
            "js": [ "content.js" ],
            "all_frames": false,
            "run_at": "document_start"
        }
    ]
}

Content Script:

let ytmusicPlayerBar

let waitForPlayerBarInterval = setInterval(() => {

    if (document.querySelector("ytmusic-player-bar")) {
        ytmusicPlayerBar = document.querySelector("ytmusic-player-bar")
        clearInterval(waitForPlayerBarInterval)
        getInfo()
    }

}, 1000)

function getInfo() {

    let getInfoInterval = setInterval(() => {

        if (ytmusicPlayerBar.querySelector(".title").innerText) {

            let info = {
                song: ytmusicPlayerBar.querySelector(".title").innerText,
                artist: ytmusicPlayerBar.querySelectorAll("a")[0].innerText,
                albumName: ytmusicPlayerBar.querySelectorAll("a")[1].innerText,
                albumCover: ytmusicPlayerBar.querySelector(".image").src,
                time: ytmusicPlayerBar.querySelector(".time-info").innerText
            }
        
            console.log(info)

        } else {
            console.log("Nothing is playing.")
        }

    }, 1000)

}