It looks like there's a bug in routing floating IPs. When connecting from another droplet within the same datacenter through floating IP, the connection sometimes works and sometimes doesn't. Using droplet's own IP address immediately resolves the problem at the cost of losing floating IP functionality.

That means floating IPs are presently useless for backend servers. It also means it is impossible to publish application APIs through floating IP address, because there is always some non-trivial probability that some API client happens to be in the same datacenter.