Along with the song title you also need to check the same condition with the values of lyrics object. You can get values of lyrics object by passing object to Object.values(). And with Array.some() you can evaluate if there is any element satisfying condition. As Array.some() returns a boolean it will make it easy to check.

fix would be.

const songs = [
{
    "createdAt": {
        "seconds": 1654000036,
        "nanoseconds": 204000000
    },
    "lyrics": {
        "liryc-1-b": "Some example lyrics verse 1",
        "liryc-1-a": "Some example lyrics verse 2"
    },
    "category": "maculele",
    "title": "asdasd"
},
{
    "category": "corridos",
    "createdAt": {
        "seconds": 1653936021,
        "nanoseconds": 438000000
    },
    "lyrics": {
        "liryc-1-b": "lorem ipsum dolor lorem ipsum dolor lorem ipsum dolor lorem ipsum dolor lorem ipsum dolor lorem ipsum dolor ",
        "liryc-1-a": "lorem ipsum dolor lorem ipsum dolor lorem ipsum dolor lorem ipsum dolor lorem ipsum dolor lorem ipsum dolor lorem ipsum dolor lorem ipsum dolor lorem ipsum dolor lorem ipsum dolor "
    }
 }
]

const query = 'dolor';

songs.filter(song => song.title?.toLowerCase().includes(query.toLowerCase()) || Object.values(song.lyrics)?.some(l => l.toLowerCase().includes(query.toLowerCase()))).map(r => console.log(r));

Also, you need to check if the song title is null or undefined as a second object from songs missing title property. You can do that with Optional Chaining.

Try to filter the lyrics object keys or values using the some method:

<SongbookContainer>
  {query === ''
    ? songs.map((song, i) => <SongPreview key={i} song={song} />)
    : songs
        .filter((song) => {
          return song.title.toLowerCase().includes(query.toLowerCase()) && 
            // If you need to filter by key
            Object.keys(song.lyrics).some(lyricKey => lyricKey.toLowerCase().includes(query.toLowerCase())) 
            // Or, If you need to filter by value
            Object.values(song.lyrics).some(lyricValue => lyricValue.toLowerCase().includes(query.toLowerCase()))
        })
        .map((filteredSong, i) => <SongPreview key={i} song={filteredSong} />)}
</SongbookContainer>;

Maybe not the best answer but it's work.

const query = "title";

/**
* tools to parse data any data to string and find correspond word.
*/
function wordExist(data, word) {
  if (!data || !word) {
    return false;
  }
  const parse = JSON.stringify(data);
  return parse.includes(word);
}

// use like this in your current filter
  array.filter(
    (song) => wordExist(song.title, query) || wordExist(song?.lyrics, query)
  )
);