Understanding MITM in an example authentication protocol

Imagine that we have a protocol like this:
B -> A: R_B
A -> B: {R_B,B}_K

Goal: authenticate A to B
K: a shared key between A and B
{}_K: encrypting by K

After receiving {R_B, B}_K by B, B is able to authenticate A. But what if we have something like:
A -> C: {R_B,B}_K
C -> B: {R_B,B}_K
so in this case B will authenticate C instead of A, isn't it?

In the question's example, C is acting a Man-in-the-Middle (MitM), impersonating B with respect to A (at the communication level) when receiving {R_B,B}_K, and impersonating A with respect to B (similarly) when sending {R_B,B}_K. Absent measures not described, B will accept the message {R_B,B}_K sent by C as if it came directly from A.

Such possibility of a MitM is a standard hypothesis in cryptography and protocol design, because it's it's physically possible in the overwhelming majority of communication means in use today: wires, optical fiber, radio (including satellite).

Whether "B will authenticate C instead of A" is a matter of convention and context, and these vary.

Absent other attack¹, the protocol insures that A participated in the exchange, and I would not say that the authentication property of the protocol limited to what's described in the question² is broken or useless. For example if A is a remote control for a device B, then the remote was involved within the time period between generating R_B and accepting {R_B,B}_K. If further B verifies that this time period is as most $t$, then B has the insurance A is within distance $t\,c/2$, where $c$ is the speed of light.

¹ Attacks are possible or not depending on properties of the encryption used, and on how parties with knowledge of key K (including A and B) use it beyond what's described in the question.

² Things would be quite different if the question ended in: having authenticated A, B proceeds to communicate.

The main problem in this question is understanding the authentication correctly.
If B wants to authenticate A, it means B must receive adequate evidence of message authentication and entity authentication from A.

A -> C: {RB,B}K
C -> B: {RB,B}K 

In this scheme, on one hand, C is just acting like a wire, and on the other hand, B is receiving enough evidence of message authentication and entity authentication from A.