The moment of inertia is generally used as a function to convert between torque and angular velocity. Because there is only one plane through which objects can rotate in 2D, the torque and the angular velocity will always be in the same plane, and thus the function which maps torque to angular velocity takes only one input and produces only one output. This function also happens to always be a linear mapping, which means the function can be represented as a matrix. A matrix which maps just a single input value to a single output value is a 1x1 matrix, and effectively becomes just a scalar. So yes, in 2D the moment of inertia, like the mass, is a value that scales an input. The mass tells you how difficult something is to translate and scales down linear forces, and the moment of inertia tells you how difficult something is to rotate and scales down angular torques.
For a single point, the amount of work it takes to rotate it is proportional to its mass and the square of its distance from the axis of rotation.
I = m*r^2
For any other shape, you can break it down into an integral over the set of points that make up the shape. Integrals really just sum up the contributions of all of the points, distributing an equal infinitesimal part of the mass of the whole to each one. For instance, given a rectangle with a mass m, height h, and width w, you can break it up into an infinite number of micro-areas that are a width of dx, a height of dy, and have a mass of m * (dx*dy) / (w*h), i.e. the percentage of mass that little piece has relative to the area of the whole. Each piece would then contribute that area times the distance of that piece squared from the axis of rotation (usually the center of mass of the shape). I won't go into all of the details here, but there are plenty of resources online for how to compute a double integral, which is what you need for summing up contributions over a 2D area. I recommend starting here:
https://betterexplained.com/articles/a-calculus-analogy-integrals-as-multiplication/
Wikipedia has a list of moments of inertia for some common shapes here:
https://en.wikipedia.org/wiki/List_of_moments_of_inertia