Black body radiation

This is a difficult concept to talk about without using a few proper definitions. Unfortunately these radiometric definitions all sound very similar and have similar meanings but important differences.

Intensity is the rate of energy transfer per unit area. Intensity does not have direction and thus we cannot even talk about radiation having 'different' intensities in different directions by definition.

The radiant intensity, however, is the amount of energy transfer per unit solid angle.

The radiance is the amount of energy transfer per unit area per unit solid angle.

Lambert's cosine law states that the radiant intensity of an ideally diffuse emitter (e.g. a perfect black body) is proportional to $\cos^2\theta$, where $\theta$ is the angle from surface normal.

However, the apparent surface area of a flat object is also reduced by a factor of $\cos^2\theta$ when viewed from an angle $\theta$ from normal. Thus the radiance does not vary with angle.

When you use the term intensity in your question, you probably mean radiance, which is the most natural term to talk about. It is the same as talking about how bright an object appears. To quote wikipedia:

Radiance is useful because it indicates how much of the power emitted by an emitting or reflecting surface will be received by an optical system looking at the surface from some angle of view.

In summary:

Is the radiance the same in every direction? Yes.

Does the radiance in each direction have the same spectral distribution? Yes.