The C is the C. That it is at 8, 16, 32, 64, 128 or all the bits you like. It is no coincidence that the language includes the keyword sizeof which allows you to know the size of memory reserved for all the types at run time, therefore the limits of information storage. Without considering that you have the limits.h file which provides you with a large set of constants that define dimensions and limit values for the compiler you are using. What should be avoided would be the use of non-standard libraries that you may not have in other compilers.
But even in this case with the help of directives to the preprocessor you can operate several branches of code in compilation so that your source works on different platforms.
It is hard work, but someone has to do it 😋
If what he means by 16-bit is DOS mode then no, you probably shouldn't be learning that.
Other than that, 16-bit only signifies the architecture word size, although not one which is supported by modern operating systems (although there are still many embedded devices with 16-bit processors).
Vijay Kumar c language has been around 50 years. Yes learn c, cpp, and c#.
GeeksforGeeks has a very good explanation why to learn c:
"Being a middle-level language, C reduces the gap between the low-level and high-level languages. It can be used for writing operating systems as well as doing application level programming. Helps to understand the fundamentals of Computer Theories."