Currently it's only possible to add each smarthome device to a single group.
This poses the user experience problem of not functioning in a natural way for users. Most people tend to think of their items as fitting into multiple groups, imagine a Venn diagram of your home for example.
The simplest example I can think of is a "All Lights" group. It's the first thing I set up when I found groups as a smart home option. This gives me the option of saying "Alexa, turn off All Lights" and I know every light in the house will turn off but not any other smart devices.
The problem arises when you want to then turn of all lights in the bedroom for example. At that point the most elegant solution is to add a subset of lights to a "Bedroom Lights" group, thus allowing the command "Alexa, turn off Bedroom Lights". However, this isn't currently possible as each smart connected device can only be part of one group.
(There is a hacky solution to this, using a routine to listen for the command "turn off bedroom lights" and then adding an off command for each light in the bedroom to the routine but it's hardly straightforward and you'd have to do this for every room in the house instead of just managing your groups as you get new devices).
Overall, I'd like to present the feature request that each smart device be allowed to exist in multiple groups.