They need not always represent the same physical content.
In general, if the dimensions are same, the quantities do represent the same physical content.
Like work and energy have the same dimensions and represent inter-convertible quantity.
For example :
1) Energy or work done has the units : Newtons-meters
dimensions : M L² T⁻²
Moment of a force = couple or torque = Newtons -meters
dimensions = M L² T⁻²
They perhaps do not represent the same physical content.
2) Force constant k of a spring and surface tension on a liquid surface
Surface tension = surface energy / area = force / unit length = newtons/meter
= M T⁻²
Force constant of a spring k = Newtons/meter = M T⁻²
Perhaps they are not representing the same quantity.
3) Pressure = Newtons/m² = M L⁻¹ T⁻²
stress = Force/area = M L⁻¹ T⁻²
These two quantities represent the same quantity - same meaning and content.
Modulus of elasticity = stress / strain
Since, strain does not have any units or dimensions, Modulus has same dimensions as stress. But, the content or meaning or range of values of modulus are quite different from those of stress.
4) there are many physical quantities with no dimensions ie., M° L° T° -- refractive index, Avogadro number etc. but their meanings are quite different.