What integrated circuit has the capability to control the connection between two I/O pins for signal transmission
Formally this is called a "crossbar switch", which is an actual category of parts you can buy.

The most applicable general-purpose parts to improvise one from would be CPLDs and FPGAs, though you'd be seriously underusing the capability of the latter.

An MCU is generally NOT a good choice for this task except in low speed circumstances, as MCU I/Os could only be universally bridged by the involvement of software, which introduces latency, sample rate limitations, and likely sampling jitter.  The exception might be certain MCU's that have a degree of almost FPGA-like arbitrary I/O pin mapping, though that is generally to put any functional block on any pin - any ability to directly map one I/O pin to another would be a sort of side effect.