Where I work, we call these types of requirements Cross Functional Requirements (CFRs), and their importance holistically is very significant.
Generally our approach to CFRs is to build them in as a part of the acceptance criteria of our stories. For example, if accessibility is an important CFR for your product, we ensure that every story describes how it needs to be implemented with regards to accessibility.
Pat Kua also has a handy blog post about other ways to handle CFRs
We usually verify that we're successfully implementing CFRs through automated testing. Though that's not always possible, usability is a good example of a situation where user research and practical testing replace automation.
The important thing to realise with CFRs is that not all of them matter equally, and it's important to determine which CFRs do matter to your team/product/customers to avoid spreading yourself too thin trying to make a perfectly secure, super performant, brilliantly usable, perfectly localised, and hyper accessible application when actually all it needed to be was scalable 😜
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