different type variables with same identifier linked together

The declaration int x; in a.c and char x; in b.c are only tentative definitions of identifier x.

The C11 standard draft N1570 states:

6.9.2 External object definitions
...
2 A declaration of an identifier for an object that has file scope without an initializer, and without a storage-class specifier or with the storage-class specifier static, constitutes a tentative definition.

If instead you initialize x in both the files (something like int x = 2; in a.c and char x = '1'; in b.c, they become "complete" definitions and then you will have multiple definition error from linker.

Something like:

Error   LNK1169 one or more multiply defined symbols found  
Error   LNK2005 x already defined in a.obj  

Introduction

The C standard does not define the behavior of defining an identifier with external linkage twice. Some behavior is commonly defined as an extension to C, notably on Unix systems. However, this extension relies on the definitions having compatible types; the result of defining int x; and char x; is generally not defined.

Discussion

Defining an identifier with external linkage twice violates a constraint in the C standard, in C 2018 6.9 5 (bold added):

If an identifier declared with external linkage is used in an expression (other than as part of the operand of a sizeof or _Alignof operator whose result is an integer constant), somewhere in the entire program there shall be exactly one external definition for the identifier; otherwise, there shall be no more than one.

In your program, x is used in the expression &x, so the above constraint applies: There must be exactly one external definition for x. When a constraint is violated, the resulting behavior is not defined by the C standard, per C 2018 4 2.

Why then does int x; and char x; behave differently from int x = 0; and char x = 0;? One might think they should be the same because the former are tentative definitions (because they have no storage-class specifier or initializer) and C 2018 6.9.2 2 says:

If a translation unit contains one or more tentative definitions for an identifier, and the translation unit contains no external definition for that identifier, then the behavior is exactly as if the translation unit contains a file scope declaration of that identifier, with the composite type as of the end of the translation unit, with an initializer equal to 0.

There are two reasons. The first is the rule about violating a constraint resulting in behavior not defined by the C standard is an overriding rule; it takes priority over the rule about tentative definitions.

The second is that, although the C standard does not define the behavior, other documents may define it. As noted in C 2018 J.5.11 (which is an informative section rather than a normative part of the standard), a common extension to the C language is to permit multiple external definitions. Generally, the types of the definitions should agree, and only one should be initialized.

For example, the Systems V Application Binary Interface describes how multiple definitions may be reconciled in cases where there are mixed strong and weak definitions or there are mixed common and non-common definitions. The compiler cooperates with this extension to C by producing an object file that marks identifiers differently according to whether they have regular definitions or just tentative definitions. For example, compiling a file containing char x; with Apple LLVM 10.0.0 and clang-1000.11.45.5 for x86_64 produces a symbol x marked for the common section, but compiling a file containing int x = 0; produces a symbol x marked for a general section. (When the nm command is applied to the object file produced by the compiler, it shows C and S for these sections, respectively.)

Summary

The result is:

• Defining x twice is not defined by the C standard.
• The compiler and linker extend the C standard to allow multiple tentative definitions of x along with at most one regular definition.
• In spite of the extension, the behavior of defining x with int in one place and char in another place is improper but is not diagnosed by the linker.