When it comes to using Twitch embeds on your website, Twitch’s Terms of Service and Community Guidelines are important to consider. According to Twitch’s terms of service, you are allowed to embed Twitch streams on your website as long as you comply with their guidelines, which include giving proper attribution to the original streamer and not altering the content of the stream in any way.
Regarding copyrighted music, if the streamer plays copyrighted music during their stream, they may be liable for copyright infringement. However, it is unclear whether the website embedding the Twitch stream could also be held liable for copyright infringement. It is recommended to consult a legal expert to get a better understanding of the legal implications of embedding Twitch streams on your website, particularly with regards to copyrighted material.
In summary, while Twitch allows embedding of their streams on external websites, it is important to comply with their guidelines and consider potential copyright issues.