Detect a sequence inside a bit stream
I think the misunderstanding comes from the "4-bit serial-in parallel-out shift register" term. This can be interpreted in two ways:

Deserializer: every 4 clocks it outputs 4 parallel bits.

Shift register: it outputs 4 bits every clock, of which 3 are the shifted output bits from the previous clock cycle, and one is the new bit.

The HC164 which is labeled "serial in parallel out" register behaves like the latter and that was probably what you were thinking about. This is the circuit you need.

If your teacher interpreted it as the former, as a deserializer, then that wouldn't work since it would only detect the pattern if it was aligned on a 4-bit boundary.

Note you don't need a comparator, which usually computes "greater" "lower" and "equal" since you just need an "equal" output signal. As shown by Marcus Muller, XOR gates are enough.

Note the output signal should be registered with a flop. As the register shifts, the output signal will probably glitch so it has to be sampled by a flop after it settles.