Is it true to say that both a "shared server environment " and a "VPS environment" can be either IaaS or PaaS? Say I rent a DigitalOcean 20$ "VPS environment" (IaaS) or a SiteGround 10$ "shared-hosting environment" (PaaS). 

As far as I know, in both cases I share hardware with other customers (both constellations aren't "dedicated" - either dedicated "on-perm" or dedicated in any other hosting model).

From what I just described, I get the impression that the notion of a "shared hosting environment versus a VPS environment" is misleading because in both cases there is direct sharing of one system (one hardware system - "shared" or "VPS"). 

Hence I ask, is it true to say that both a "shared server environment " and a "VPS environment" can be either IaaS or PaaS?
Usually, Only a VPS hosting can be either.

While an IaaS is always VPS, what's commonly called "shared hosting plan" is PaaS in which a client shares the same hardware and webserver software with different customers (due to sharing a webserver, it's not a VPS --- Virtual Private Server).

In a PaaS (of the subtype Pass;VPS) only the hardware is shared (so it's indeed a "shared server", but only in that sense).