Hi vtmoz
It's funny you should say "so they started calling themselves on the internet 'Moz SEO' " because Moz used to be called SEOMOZ and they changed to just MOZ so that they could concentrate on building the brand. There is evidence to suggest that if an article is written about a subject and your brand is linked from that article that Google will recognise the contextuality of the article and associate your brand name with that subject.
For example, one would expect that Nike and 'sports', 'sports clothing' etc would occupy the same space. In the same way - if your brand sold Tennis Rackets, for example, Google would begin to associate your brand name with that range of products.
In the same way, when we SEOs are optimising a page we will use Keyword Explorer. The tool would give us contextually relevant keywords to use alongside a brand name in order to strengthen the SEO of a page.
So if I were writing an article on Nike and I hadn't used the words 'sports', 'sports clothing', 'sports footwear' or even 'Phil Knight', then MOZ will suggest these as relevant keywords to use to strengthen the page's SEO.
I wouldn't include them in the brand name but I would use them on a page. In this way, they become kind of 'baby anchor text' in that they surround the brand.
In order to strengthen that further, you might write an article that is submitted to a blog with a specific anchor text backlink which is not your brand name but the most relevant keyword. So if Nike wanted (and it does of course) to be found for the term 'Sports Footwear' then it would link back to its own page with that anchor text.
Very often when I am constructing Meta titles and heading for a brand page then I would use the subject after the brand as that is often the way someone may search for it in Google.
Example: I would title a page 'Clarks Boots' as the opening two words of the title and as the H1 on the page if I wanted that page to rank for the brand and the category. I would then fill the page with boot products along with a good 300 word + description of the types of Clarks boots, the gender, heels, ankle, mid, high and any other contextually relevant keywords.
I hope that goes someway to answering your question,
Regards
Nigel