If you use an Adobe product do you really care about the Adobe logo?
Not directly, but the brand as a whole is important in helping to convey their place, their direction, their goals. Everything about Adobe’s software and user interface is ‘Adobe’. It is not neutrally a functional interface it is part of who they are and designed accordingly.
The logo, in and of itself, as a stand alone graphic has little immediate, direct impact, but as part of a wider whole, it is definitely of some import. A logo is only ever a mnemonic for an organisation’s culture, ethos and aspirations. If not, why is Adobe’s brand identity different to Affinity's, or Quark’s? They all do different jobs in representing those companies, services and products and in helping us make our choices. Adobe is an odd one, in that, it is slightly different because it pretty-much has the market sewn-up and has a captive audience amongst pro designers.
I understand what you are saying, given the UI/UX nature of this particular case, the emphasis is far more leaning that way, but I always think it is a flawed approach to ever split them up and treat them separate, individual components. They are symbiotic and designed correctly, work with each other to support each other. Even if you are designing ‘just’ a UI, it is always a UI for a company, or organisation of some description and should sit within their ‘culture’ and be utilised for a particular purpose.