Yesterday I confirmed it on the EU PC/Mac server. I had only one character, and had opted for the free room in Daggerfall. I haven't bought any other housing because I've been working on maxing out my inventory and bank upgrades. Yesterday I finally bought the last bank upgrade, which means my next in-game purchases will be to start collecting the player housing, but I didn't want to buy the rooms at the inns if I could get them for free.
So I created a new Argonian character and, upon leaving the mind prison, took the Shimmerene Wayshrine to Stonefalls, made my way to Ebonheart, and acquired the Ebony Flask Inn Room for free.
Then I created a new Khajiit character, took the Shimmerene Wayshrine to Vulkhel Guard in Auridon, and acquired the Mara's Kiss Inn Room for free.
Next, I created a new Dark Elf character, took the Shimmerene Wayshrine to Seyda Neen in Morrowind, strolled over to Vivec City, and acquired the Saint Delyn Penthouse for free.
Finally, I created a new High Elf character, headed west across Summerset Isle to Alinor, and acquired the Golden Gryphon Garret for free.
I just wish I'd thought to do this on the NA server instead of spending a total of 12,000 gold to acquire the other four rooms. Of course, it doesn't take me very long for my main NA server character to scrape up 12,000 gold from thieving and crafting-- but it did when he was still a low-level character.
Counting the Psijic manor, I've now acquired six housings for free on the EU server. And since Elsweyr is going to be a "Chapter" like Morrowind and Summerset, I'm guessing it will also have a room at an inn that can be acquired for free.

