Installing Applications in Isolated Environments Is it possible to install (multiple) instances of the same app in an isolated environment, whereby it can't detect other apps, or read/write files anywhere else?
Yes just as the other answers have suggested. AFAICT, this mechanism is now being implemented in newer ROM versions by some manufacturers, and even some independent developers.

For example, Lenovo has incorporated a similar isolation mechanism in order to allow installation of multiple instances of applications or at least dual installation. 

This feature is called Dual Apps in some Lenovo devices:



The brief description states:


  Dual Apps will let you run two separate instances of the same app.
  Both instances will run independently of each other and will not share
  any kind of data with each other, except for contacts provided you
  have allowed that.


I believe this idea has been inspired with the recent surge of dual sim-powered smartphones and growing popularity of social networking applications such as WhatsApp, Facebook etc.

For this reason Huawei's EMUI 5.0 has implemented the same feature called App Twin 

 

Essentially for now it supports WhatsApp and Facebook but more applications can be added as suggested on this XDA thread.

However it seems its not as fine-tuned as the former as it has some limitations :


  You cannot backup or clear App Twin account data separately. When you
  enable an app in App Twin and perform a backup, both the main account
  and twin account are backed up. When you clear app data, both accounts
  will be deleted.




Probably the most interesting project on this topic now is Island

The developer (Oasis Feng) is working on a beta version of this project, and is described


  Island is a sandbox environment to clone selected apps and isolate them from accessing your personal data outside the sandbox (including
  call logs, contacts, photos and etc) even if related permissions are
  granted. Device-bound data is still accessible (SMS, IMEI and etc).
  Isolated app can be frozen on demand, with launcher icon vanish and
  its background behaviors completely blocked.


The features are summarised as follows:


app freezing, 
privacy protection,
parallel accounts


As mentioned above its still in beta, and installation is via
Opt-in the open beta test on Google Play