As per comment#0 considering this as Feature Request and marking it as Untriaged for further inputs.

[Extensions triage] : Can you elaborate on your use case. Generally, I'd imagine that one would want to run a subset of extensions in the incognito mode (as compared to the normal mode).

Having the ability to enable apps/extensions only incognito mode may turn to be a useful tradeoff for the user on an ephemeral basis during each incognito session. 

It is common for apps/extension to demand various data permissions pertaining user's overall browser interaction, some users are very sensitive to provide that level of access all of the time to third-party code. 

So basically, testing environment for untrusted extensions? That sounds useful, but note that you could achieve that today simply using a separate profile.

To answer and clarify for comment#5, this feature request suggests an ability for a user to install and use extension/apps only in Private-browsing mode. 

It is assumed that such an ability allows the users to allow extensions/apps the required data access permissions only on ephemeral basis avoiding the overheads of an additional browser profile management. 

For another perspective, such a feature may also be thought useful where users would want to mitigate privacy risks for one or more least trusted extension/apps being enabled only in the Incognito mode.