Companion Restrooms are single stall restrooms with a door. They are set up for use by people with disabilities and have raised seat toilets, roll under sinks, grab bars and room by the toilet to park a wheelchair for transfer.
Handicapped bathroom stalls are simply stalls in a restroom set up for use by handicapped people. There are 2 types - one is set up similar to Companion Restrooms and the other has only a toilet with grab bars and no room for a wheelchair.
In some places, the Family Restrooms have extra things, like child size toilets and changing tables. In others, they are set up exactly like WDW's Companion Restrooms. They are actually Companion Restrooms, although they are not labeled that way. Labeling them as 'Family Restrooms' causes problems for 'bathroom police' who see someone with a disability using them and are upset because they are labeled 'Family'. That is one reason WDW uses a different label.
NONE of the WDW ones are set up with child size toilets and many do not have changing tables.
WDW has them labeled as Companion Restrooms so people are not expecting those 'family restroom' items. Their location was only listed in the WDW maps/literature for guests with disabilities because there are do few of them in each park and they are the only toilets in the whole park that many people with various disabilities (not just mobility) can use, especially if they need assistance from someone of the opposite sex.