That is an insightful question - there are similarities and differences. It is important to understand what each is and why they exist.

Context switching is the process of switching a process from working on one task to working on another even before the former task is completed. This involves saving the state of all volatile data like registers, program counter, memory, etc. (in other words the "context")  to persistent storage and then loading up the context of a new process. Or just starting up a new process from scratch which will have its own context. This technique allows a process to concurrently work on multiple tasks without the necessity of requiring one to run to completion. 

Interrupt handling is the process of a processor or process receiving an asynchronous signal that typically requires it to "drop" what ever it is doing and work on this interrupt immediately.  By definition it requires a context switch to accomplish this. Once the interrupt is serviced the earlier task can resume. Interrupts can be hardware (which seem to be what you are referring to) or software (like UNIX signals). 

So Interrupt handling does require a context switch and your real question probably was - are the terms interchangeable?  No, because you can have a t context switch that is not driven by an asynchronous unpredictable interrupt.  The most obvious use would be in a multi-tasked operating system that allows for multiple processes to run concurrently on the same processor (not to be confused with parallel processing on different processors). To do this the OS needs the ability to execute part of a process, save its state and then execute another process. In a time shared operating system this would be done by giving a fixed time slice shared in a round robin between processes. There are other ways too, based on condition synchronization etc.

So in a nutshell, interrupt handling is just one example of why you would need context switching. As a few minor comments on your question, it is unlikely processor state would be saved in a stack, it would probably be on a heap. Because the pattern does not have to be LIFO. Also a controller very much has an operating system. It just isn't a big famous operating system like Windows or Linux. By definition it has an operating system because that is what is doing the context switching.  It just is a relatively simple embedded operating system.