How to measure/display the delay of two cosine signals on the oscilloscope?
This is a MATLAB problem not an electronics problem. You are wanting to play two channels of audio using soundsc. However you are supplying it a row vector instead of a \$x \in \mathbf{R}^n\$ i.e. an \$n \times 1\$ vector.

If you have two vectors that are \$m \times 1\$ and concatenate them, you end up with a \$2m \times 1\$ vector, whereas if you want to play things simultaneously you want a \$m \times 2\$ matrix. What you will get is a 2 second signal followed by a click caused by a discontinuity as you switch between the original signal and the phase shifted signal, followed by another 2 seconds of sine.

Here is the fixed source code, where I have change your comma to a semicolon in the last line to create an \$m \times 2 \$ matrix.

clear all
close all
%% Create sinusoidal Wave
fs = 44100;
t = 0:1/fs:2; T =10e-3;
signal = cos(2*pi*t*1/T);
%% Create shifted signal
Delay = 5e-3;
shifted_signal = cos(2*pi*(t-Delay)*1/T);
%% Output signals
soundsc([signal; shifted_signal],fs)