The short answer is: no, they are not.
CX (customer experience) focuses on the more strategic direction of design in terms of managing the many different channels that links the enterprise/organization with the end-user.

While service design does emphasize the need to think holistically and approach solutions with an end-to-end perspective, at the end of the day: it is about designing, optimizing, and researching service experiences. CX is about infrastructure. It is the totality of the all touch-points across the many different channels and mediums and establishing a strategy for each one. As an example:

01. Channel A - Website: strategy is to showcase the value of our offerings
02. Channel B - Social Platforms: strategy is to tell a story about our brand
03. Channel C - Customer Support: strategy is to make customer feel valued
04. Channel D - In-Store Retail: strategy is to bring our products to life
05. Channel E - E-mail Comms: strategy is to make customer feel informed

Once each channel has a defined strategy (or ideal experience / outcome) established, the CX professionals work with cross-functional teams—including UX and Service Design—to set the proper strategic priorities centered around experience. Executed successfully, these priorities aligned with business goals or projects will demonstrate positive return on investment in the form of metrics such as:
• Net Promoter Score (NPS Survey)
• CSAT (Customer Satisfaction)
• Customer Effort Score (CES Survey)
• Churn Rate (% of customers that walk away from product or service)
• Retention Rate (% of customers that stick with your product or service)

It’s clear to see that CX has to work closely with business executives to determine, for the organization, where the priorities have to be to achieve customer excellence. Among the strategic-oriented questions a CX professional or team may ask is:

-What does it mean exactly to serve our customers?
-In which ways do we deliver value for our customers?
-What experiences are most impactful to delight our customers?
-Do our current business process support our mission for customer excellence?

Service Design focuses more on the execution side of things. Based on the strategic priorities set forth by the “CX Department”, the Service Design team(s) have a guiding direction when approaching specific projects or tasks. For example, if Service Designers are working on improving aspects of the Customer Support "service” experience, they understand that the guiding principle is to make the customer feel heard and valued. They are designing and conceptualizing within the context of a larger CX strategy framework.

In summary, we can think of a CX practice as the bridge between corporate executives with high level business goals and the varying teams and functions that support these goals (design teams included). Once a company decides that customer experience is a core strategic priority, they will want to hire a CX Officer, VP, or Director to articulate specific outcomes and objectives that will translate this priority into tangible results (KIPs, Metrics, etc.). Therefore, CX as a practice is less about design and more about strategy & brand execution; however, it is closely connected with all the design teams to execute “experiences” across channels and touch-points. However, CX professionals are also in the practice of evolving business processes and protocols to achieve the desired results. Again, only at an executive level (or close to it), can these kind of decisions be implemented to improve the systems in place so the organization is delivering value at every customer interaction = how are you making their life easier, simpler, faster?

CX is about infrastructure.