If you want to join a mailing list provided by Google Groups and just happen to land on the Google Groups page, it will force you to either login to your Google account or if already logged in, to signup with the email associated with your Google account. If you want to use any other email address however, it seems like you can't. There is no information anywhere in Google Groups itself that seems to suggest you can do this.
I personally have a separate email address for open source related topics and a project's mailing list is often the best way to keep track of the latest developments. Luckily it's very easy to do if you know how.
First send a blank email to
{group}+subscribe@googlegroups.com. You'll receive a response
that looks like the following:
…but don't click the button, you'll arrive at the Google Group again and
nothing will have changed, it will still prompt you to confirm with your
Google account. Just reply to the email (which sends it to
{group}+subconfirm@googlegroups.com). You'll receive a
further email stating you have successfully joined the group.
That's it. To unsubscribe, it's the exact same process but the initial
email needs to be sent to
{group}+unsubscribe@googlegroups.com and the response will be
sent to {group}+unsubconfirm@googlegroups.com.