The quantities which have the same dimensions as those of solid angle are:

Solid angle ($\Omega$) is a dimensionless quantity $[M^0{L}^0{T}^0]$.Angle ($\theta$): Dimensionless $[M^0{L}^0{T}^0]$.Strain: $\frac{\Delta L}{L}$, which is dimensionless $[M^0{L}^0{T}^0]$.Stress: $\frac{F}{A}$, has dimensions $[M{L}^{-1}{T}^{-2}]$.Arc: Length, has dimension $[L]$.Angular speed: $[\theta /t]=[T^{-1}]$.Both strain and angle are dimensionless, just like solid angle.