Yes, you need to tell Let's Encrypt, when asking for the certificate, every name which should be listed on that certificate. It is also OK to ask Let's Encrypt for more than one certificate (but again, if you need lots please check the rate limits) This will work like that middle option on the page you linked, UCC/SAN SSL. SAN stands for "Subject Alternative Name" which is a way for a certificate to list more than one name, and for some reason Godaddy limits it to just five (maybe they charge extra for more?) but as leader wrote, Let's Encrypt will allow up to 100 names per certificate and they do not charge (although they do accept donations ).
Any public CA needs to check that you really control all the names you've asked for - so that you can't obtain a certificate for somebody else's site. The "certbot" software and numerous other clients for Let's Encrypt will help you prove this to Let's Encrypt automatically, so that hopefully you just set it up once and it gets renewed certificates every few months as they're needed. The most popular method is by placing a file on the web server, in a special place, so that Let's Encrypt can fetch that file - but there are other methods if that doesn't help you. If you use a web hosting company you might even find that they'll sort all this out for you, and you can just click to say "Yes" you want a free certificate for your sites in their control panel and you're done.