not exactly that way. In ID you can create object style (Window>Styles>Object Styles palette), apply it to objects. Aftewards, when you change style attributes, all objects with that style applied wil be updated accordingly. Much similar like text styles or color swatches.
However, changes are limited to attributes shown in Object Style Options box.
Also you should be aware of a Clear Overrides When Applying Style option, found in Object Styles palette's menu.
ID does not share Illustrators sybol logic, but there are a couple of possibilities that might work for you. For CS4 and later you can create the content you want to "clone" as a separate file, then place that file as a link as many times as you like in other files. Update the original, then update the links.
In CS6 you can "Place and Link" a selected object object using the Edit menu. That's about as close as you get to Illy's behavior. In CS5.5 you can place and link a story, but not a graphics frame.