For just about any shape of cut-out, one could use a router bit in a router, but they are typically quite heavy and more powerful than you need for cardboard. More appropriate would be a tool commonly called a Dremel tool. These are rotary multi-feature tools for which one can purchase different attachments and cutting bits.
A router head for such a tool is going to be smaller than a full size router. Cutting bits are typically around the 1/8"/3 mm size. There are larger diameters but not as much variety as with a full size router.
The tools are often variable speed and some are cordless, offering even more flexibility for use.
For cutting straight lines, one would want to have a guiding device such as a flat board clamped or taped to the cutting surface. For curves and holes, it's going to be mostly eyeball guidance.
If you are cutting only straight lines, there is available what amounts to a small scale hand-held circular saw, but that loses the ability to change kerf sizes.
The one shown is again by Dremel, but other manufacturers have similar products.
Last on the list would be an oscillating cutter.
Many manufacturers, corded and cordless, with multiple tool selections. No kerf adjustment, generally speaking, more easily controlled than a circular cutter but slower.
I did not suggest a hobby level CNC cutter as they are expensive and limited to the bed size, which limits your work piece size as well.
