Why is torque not measured in Joules?

Torque is force at a distance. Work is force through a distance. Same unit dimensions, different measurements.

A joule is defined as a specific amount of energy or work done. Torque is neither one of those, so even though the units are the same the meaning of joule cannot be applied in the case of torque.

Joule and Newton meter are two units that are algebraically identical; you might say they are two names for the same unit. This is not the only example: Ohms is a unit of resistance, while "ohms per square" is an algebraically identical unit of sheet resistance. Hertz is a unit of frequency, becquerel is a unit of frequency in the context of radioactivity. In Gaussian units there is a delightful example of five algebraically identical units.

Why do people use different names for the same unit? Just the simple reason: It facilitates communication and avoids misunderstandings. If I mumble something and point and say "50 newton meters", you can be pretty sure I'm talking about a torque; if I say "50 joules" you can be pretty sure I'm talking about an energy. Therefore, having these different terms helps reduce the frequency of communication mistakes (albeit only to a limited extent).

The fact that torque and energy have algebraically identical units does not mean torque and energy are the same; in fact, it means nothing whatsoever. Torque and energy are completely different concepts that just happen to have algebraically-identical units. (Well, I suppose torque and energy are connected in various ways, just as any two randomly-selected quantities in classical mechanics are connected in various ways.)

It is not uncommon for units of a different physical entity to be used to measure a related physical entity. e.g. distance is generally measured in meters; but it is also measured in light years which is the distance traveled by light in a year. The important thing is that there should be a consistent way to convert one unit to another.

Someone pointed out that Torque is a vector (defined as a cross product) while Work is a scalar (defined as a dot product). However, that can't be "the (only) reason" for different units. Units are defined for "magnitude of a vector", which by itself is a scalar. So, the reason you can't use Joules for torque is because there is no consistent way of converting Newton-meters to Joules and vice versa.

There are 2 types of units viz., the basic/elementary units for mass, distance and time and the compound/derived units such Newton, Joule, etc for physical phenomenon that are derived from the basic units.

So, 1 Newton is the amount of Force required to increase the velocity of 1 Kg of point mass by 1 m/sec in 1 sec, in the direction of the change in velocity. 1 Joule is the amount of work done when a force of 1 Newton moves any point mass by a distance of 1m.

For a unit of Joule to be used for a unit of Torque, you would need a unit of Torque to always perform 1 Joule of work, which is not true.

The reason we distinguish the two is that torque is vector quantity, where as energy is a scalar quantity. So while we give the magnitude of torque the same units as energy, there is in fact additional information that tells us the direction the torque is applied.

UPDATE: As dmckee has pointed out in the comments, to be perfectly corrected torque is a pseudovector, which is equivalent to a mathematical bivector in three dimensions. This distinguishes it from a true polar vector. The distinction is important since the dimension of the pseudovector is n-1 instead of n. This is important conceptually as it is critical to our understanding of conservative forces and central forces, and more specifically the conservation of angular momentum.

In particular, angular momentum conservation implies that motion under central forces will always be confined to a plane.

Yes, torque has units of joules in SI. But it's more accurate and less misleading to call it joules per radian.

Let's take the simple case of a single force acting perpendicular to position (reference) vector: $$tau=rF$$ To tease out energy from this equation, let's consider an infinitesimal change in (rotational) energy $dE$: $$dE=tau,dtheta=rF,dtheta,$$ or $$tau=frac{dE}{dtheta}.$$ From this equation, one can interpret torque as the amount of rotational energy gained per radian of rotation. In other words, joules per radian in SI units. But, since one usually considers radians as unitless, this "simplifies" to just joules.

Radius is usually measured in [m], but for rotational movement it's unit is different to length namely [m/rad]. Hence the unit for torque is [Nm/rad]. Torque times angle will come out as energy. I do not know why radians are omitted, causing confusion for the understanding population.

Torque could be measured in joules per radian. Torque by angle gives energy.

$$W = τ, θRightarrow τ = W/θ$$ So the units of $τ$ must be Joule/Radian. In the SI, since the radian is dimensionless quantity, the units are dimensionally the same, but they are technically different units.