It is given that,

The diameter of the cylinder $=$ the height of the cylinder

$⇒h=2r$, where h – height of the cylinder and r – radius of the cylinder

As we know that,

Volume of a cylinder $=\pi {{r}^{2}}h$

Then, volume of the cylindrical vessel $=\pi {{r}^{2}}2r=2\pi {{r}^{3}}$….. (i)

Now,

Volume of each identical vessel

Diameter $=42cm$, so the radius  $21cm$

Height $=21cm$

Then, the volume of two identical vessels $=2\times \pi {{21}^{2}}\times 21$ ….. (ii)

As, the volumes on equation (i) and (ii) are equal

Equating both the equations, we have

$2\pi {{r}^{3}}=2\times \pi {{21}^{2}}\times 21$

${{r}^{3}}={{\left( 21 \right)}^{3}}$

$r=21cm$

So, $d=42cm$

Therefore, the diameter of the cylindrical vessel is $42cm$.