Some extensions, like TimescaleDB, have licenses explicitly forbidding their use in hosted products which compete with the companies backing them. In concrete terms, this means they can't be used in any of the major cloud providers' managed database solutions (e.g AWS RDS).
Others, like Citus, are licensed under the AGPL, which is basically an even stronger version of the GPL that requires anyone hosting a service to provide their modifications to the source code. While this doesn't strictly rule out use in e.g. RDS it does basically say that Amazon would be required to open source any changes they make to get it working in RDS, which is a burden they don't seem interested in taking on, and which might reveal details about how their services work under the hood which they are generally fairly cagey about doing.
The AGPL is also viewed even more suspiciously than the GPL by corporate lawyers who fear it may lead to virally forcing the company to reveal all of its source code. This fear is probably unfounded, but all copyleft licenses are primarily about expanding end-user freedom and so are treated as undesirable in most businesses that make money from selling software.
I asked about this in their chat 2 years ago, and the CEO responded directly. His response I believe was that you are allowed to use Timescale on managed servers, but you're not supposed to become a Timescale hoster for others.
That may mean that RDS could provide Timescale as an available extension without promoting it directly. However, that is a sufficiently gray legal area that I doubt Amazon wants to wade into it (instead of just making a competing product).
The only way I think one could safely thread the needle here would be for RDS to allow you to bring your own extensions, as then Amazon is definitely not hosting it for you. However, that would compromise the security model and responsibility division that RDS provides.
Licencing. Postgres has a well known, open source licence making it easy to use. Some extensions do not follow that practice making it more difficult to use.
Even if the extension has a liberal licence eg Apache or MIT nothing guarantees that the licence won't change in the future.
