Some programs are written in a mixture of C and C++
This is just a fact of life. You can compile object files from C and C++ and link them together. The result can quite reasonably be called "a C/C++ program".
But that's only the program as a whole. What about the individual compilation units?
There is a subset of C which is also a subset of C++
A program (or compilation unit) written in that subset will compile and behave the same under conformant C and C++ compilers. Such a program or file can rightly be called "a C/C++ program" or "a C/C++ file".
A partial program such as a header file may also be used in both C and C++ programs. Such header files can rightly be referred to as C/C++ headers.
Quoting Professor Bjarne Stroustrup:
Is C a subset of C++?
In the strict mathematical sense, C isn't a subset of C++. There are programs that are valid C but not valid C++ and even a few ways of writing code that has a different meaning in C and C++. However, C++ supports every programming technique supported by C. Every C program can be written in essentially the same way in C++ with the same run-time and space efficiency. It is not uncommon to be able to convert tens of thousands of lines of ANSI C to C-style C++ in a few hours. Thus, C++ is as much a superset of ANSI C as ANSI C is a superset of K&R C and much as ISO C++ is a superset of C++ as it existed in 1985.
Well written C tends to be legal C++ also. For example, every example in Kernighan & Ritchie: "The C Programming Language (2nd Edition)" is also a C++ program.
So yes there is such a thing as C/C++. It's anything which is both valid C and valid C++.
The C pre-processor is part of the C language. The C++ pre-processor is part of the C++ language
You can write a compilation unit which will compile under C or C++ and be different. For example, it might have basic functionality compiled in C but take advantage of a C++ library if compiled in C++.
If the program is essentially the same, but with additional features, it's not exactly wrong to say it's the same program. Its the same, but also different.
Most C programmers can do at least a little C++ and vice-versa
It's not unreasonable to call such a person a C/C++ programmer. Yes, they probably specialise in one, but is there anyone who is a competent C or C++ programmer who literally cannot do any of the other language? In a way, aren't they all C/C++ programmers?
There's nothing wrong with saying "C/C++". What matters is being understood
The English language is not a tool for expressing syllogisms. You can use English for logic, but only just, and with great effort.
This is because words do not naturally have exact meanings, but rather a vague cloud of denotations and connotations. What matters is if people understand what you are saying.