Defining quality
Branding, yes. If you think about the essence of (one aspect of) branding, it's to make life simple for users, who see conceptually similar thuings that look similar, are familiar and "known", etc. A number of branding truisms also apply to the work we do, in the sense we want to reach out and attract people and then guide them into best ways (if they choose). So yes, a lot in common. Not via "visual image" or graphical design, but philosophically.
Branding actually goes a lot further than that. While branding may simplify the user experience as far as choices, branding is actually how a thing exists in the mind of the user. Brand is NOT a physical image. The graphic I wished to upload that I sent you actually explains the complexity of brand. What we are talking about here is really about what Wikipedia is, and thus how it does what it does, because how Wikipedia does what it does is a reflection of the Wikipedia brand.
In truth, we cannot have a conversation about quality without starting the conversation with brand.
Because "Branding" is a word with strong connotations/meanings, it can confuse a discussion. Would you be okay using terms such as expectation and perception, which I think cover what you're talking about, rather than branding which tends to imply the visual design identity.
Actually I would prefer other people understand what the word "brand" really means rather than use expectation/perception because brand is not encompassed by either/both concepts. I am not talking about expectation and perception, I am talking about what something IS and how the IS drives the what something DOES. In this case, Wikipedia is the something. An IS is not necessarily visual. Brand is intangible but is expressed through that which is tangible, whether it be a mark/logo, or the way customer service responds to a customer or the way that a user experiences Wikipedia (not necessarily tangible but could be.) I have uploaded the graphic, if you need me to explain it, please ask.