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I couldn't find Epsilon Horologii in the star names in the database at all. 
Do you use a modded database ?
No. No modded DB.
I can open the DB using the program "DB Browser (SQLite)". In the table "Dim_KnownSystems", I applied the filter to the column "ConstellationName", so the names appear. In the attached image, the filtered content.
As you see, the "KnownSystemID"s are 36 and 384, and the names in the column "Name" are different. But in the galactic map the same name appeared.
If the "real Stars" database doesn't have duplicates (Sirius + Alpha Canis Majoris) which I'm pretty sure it doesn't, then you can decide that, as astronomers have in the past, that Epsilon Horologii is an optical double star (two stars in the same place in our sky, but different places in the 3-D universe, and name it Epsilon(2) Horologii.
Yes. Thanks, Steve!
I did so: Epsilon2 Horologii.

@Andrew.
Thank you.
In the list, the star GJ 1061 appears. It is the first Epsilon Horologii that I met.
I can't find GJ 3210. Searching online, nothing.