standard shape or size
It depends. Some modular standards also define the physical size of modular "units" to be something such as 2'x2' or 2'x4'. That way building a layout becomes easy when going to a show and pretty much any module can be substituted for another one. These are most often setup to make up a layout in a "roundy round" fashion.
Other modular standards only mandate where and how tracks meet the end plate / interface end, and what the end plates looks like (for example, Free-Mo and FREMO). These standards don't mandate the size of the module itself beyond the end plate. Because the length of modules are undetermined, modules can have curves that are any degree of curvature, not just 45, 90 or 180 degrees. As a result of this, creating a layout requires more planning and typically results in a "point to point" or "point to loop" type design.
All modular standards (to my knowledge) allow for sectional modules: a single module consisting of multiple sections with the overall combination of sections (the module) conforming to the modular specifications. (Note: different geographies can use the module and sectional terms reversed). For example a standard might state you have a single track at the middle of the end plate of 24" wide and modules are multiples of 2'x4'. Your module can be 4 sections of 2'x5' and meet the specs. For the Free-mo stamdard, your module could be double track on the middle of an end plate 26" wide, and the side can widen to 3' and 2.5" then narrow to only 12" in the middle before widening back after a 2 degree curve to a single track on the center of a 24" end plate, while in length your 4 sections are a total of 31' and 9 3/4" long.
For a home layout there is nothing preventing anyone from utilizing any of the modular standards. It's just that one modular standard might be better suited than another. Free-Mo and FREMO could be ideal standards for a home layout. They are the ultimate form of TOMA: a module typically depicts a single prototypical scene, and it's typically required to be in a sufficient state of completion in order to be incorporated in a layout at a larger regional event (NTS, Trainfest etc). These home modules, under Free-Mo or FREMO standard, can be any shape as long as the end plates meet the spec. And as Ken said above, even if your modules at home don't conform to the end plate spec (due to space limitations for example), a simple add-on section could be build to bring your non-conforming module to spec when you have the intend to bringing it to an event. When home, you'd have other add-on modules/sections that tie two or more modules together. If you'd move to a new place, you scrap the add-on modules and build new ones to tie the "prototypical scene" modules into the new space.
The above is obviously a simplified depiction of modular reality. If you want your home modules to be used in an event outside of your home, your module must meet all minimum specs of a standard. Thus not just end plates, but also track work (rail code, switch frog sizes, curves, easements, super elevation), wiring, DCC panels etc. All elements of your module must meet the minimum specs, or risk not being allowed in the layout or relegated to the end of a branch line. For home use, nobody cares how you build your modules, but for events, they do.
So if you take the extra effort to follow a standard completely for your home layout using modules, you get to reap special event rewards. You'll gain the ability to have your medium sized intermodal yard used to get 20 cars set out from a 150 car intermodal consist; your coal flood loader module gets to load a 105 coal hopper train with mid train DPUs; your grain elevator receives a 75 car unit hopper grain train; your plastic plant gets switched by NS receiving NS box cars and NS owned pellet hoppers instead of your BNSF (at) home road. All the while your ethanol plant module is now next to a massive prototypical 24' long bridge module for a weekend, with trains taking 2.5hrs+ to make a full run over the layout.
