Name resolving in http requests
First it all depend on how the name resolution subsystem of your OS is configured. If you are on Unix (you did not specify), this is governed by /etc/resolv.conf.  Here you can provide the OS with a search list: if a name has not "enough" dots (the number is configurable) then a suffix is added to retry resolution.
The library you use to do the HTTP request may not query the OS for name resolution and do its DNS resolution itself. In which case, it can only work with the information you give it (but it could as well re-use the OS /etc/resolv.conf and information in it), hence the need to use the full name.
On the contrary, the psycopg2 may use the OS resolution mechanism and hence dealing with "short" names just fine.
Both libraries should have documentation on how they handle hostnames... or otherwise you need to study their source code. I guess psycopg2 is a wrapper around the default libpq standard library, written in C if I am not mistaken, which hence certainly use the standard OS resolution process. 
I can understand the curiosity around this difference but anyway my advice is to keep short names when you type commands on the shell and equivalent (and even there it could be a problem), but always use FQDNs (Fully Qualified Domain Names) in your program and configuration files. You will avoid a lot of problems.
I am trying to a simple make a http request to a server inside my company, from a dev server. I figured out that depending on the origin / destination server, I might, or not, to be forced to use qualified name of the destination server, like srvdestination.com.company.world instead of just srvdestination.
I am ok with this, but I don't understand how come my DB connection works?
Let's say I have srvorigin. Now, to make http request, I must use qualified name srvdestination.com.company.world. However, for database connection,  the connection string with un-qualified name is enough psycopg.connect(host='srvdestination', ...) I understand that protocols are different, but how psycopg2 does to resolve the real name?