Design systems have shared ownership with UX, development, and product.
In the early stages of building a design system, most teams start out being dispersed. Dispersed teams are composed of subject matter experts that belong to other teams - designers, researchers, product owners, QA, developers. These things usually depend on the structure and organization size. The organization structure and size of the UX team will also dictate additional responsibilities within the design system team (e.g. design reviews, developer experience, on boarding process, etc.)
As the design system becomes more mature, a centralized dedicated team will allow the system to scale and support multiple applications and platforms.
With different layers of complexity, some design systems are built using other open source systems.
While some of those systems offer Theming, customizing the elements of open-source systems can inevitably happen as "out of the box" components don't always solve all business problems & use cases. An established governance process for when things should be customized or added to a system will avoid tech & design debt. I refer to the design systems built on external sources as micro-systems.
A design system built in-house will require a similar governance process that will address things such as adding new features, adapting existing components, pushing changes, and communicating them to the organization.
Design systems work as an aid for organizational problems. Inconsistencies are a symptom of a misaligned design production.
I worked with teams to set out a clear goal (what problems do we want the system to solve?) and base metrics which served as KPI's in early stages of the system.
Design system adoption can be measured for existing or future features. That means requests and tasks will need to be built into a roadmap, followed by a backlog to support that. Adopting design systems for future features is typically easier, as it doesn't require rewriting old legacy code, or revisiting forgotten code sources. To update legacy systems and code requires clear alignment with architecture and tech leads, as well as PO's and scope.
Measure the things you want to improve.
This will depend very much on each organization. Some examples from my past include sprints efficiency, code implementation, quality of documentation, etc.
process and team transparency
Questionnaires and surveys.
Interviews to dive into more specific data points.
Open office hours.
TEAM PRODUCTIVITY AND EFFICIENCY
SCRUM burndown charts.
SCRUM velocity.
Time spent during design hand-off.
Onboarding time spent for a developer.
Onboarding time spent for a designer.
code & design implementation
Running queries for commits information that involve the component library.
Code reviews.
Design reviews.