So I created a blog with Atahuapa under an existing website, and the default link to this blog is something like this: companyname.com/blog1. I'm wondering if I can change the link name to something unique, i.e. not using the "companyname" and give "blog1" a specific name. Thanks in advance!

IMHO the best way to do what you want is to get another domain name and use this install as an addon domain that many hosts allow. You will need to do some fiddling with settings and such because the path to some images will change and the site URL itself will of course be different so that will have to be changed in the WordPress settings.