The article in question messes around with the geography quite a bit. Areas that are climatically and geographically very different are handled under the same label, which is kind of nonsensical.
It is true that there some parts of Slovenia which "can receive between 2000 and 2500 mm of rainfall per year". Here is a map showing the corrected annual precipitation in Slovenia between 1971 and 2000, published by the Slovenian Ministry of the Environment. There are several regions colored in purple hues, indicating an annual precipitation of 2000 mm or more.
However, this is not really surprising if you know anything about the area. Here you find the Julian Alps and the Kamnik–Savinja Alps, alpine mountain ranges that have many peaks over 2500 m high, with Triglav, the highest peak, at a height of 2864 m. So obviously, there are areas in Slovenia with more than 2000 mm precipitation per year, but really, this is nothing of a surprise considering that you're in the mountains (incidentally, at this altitude, much of the precipitation will be snow, not rainfall, as the article claims).
And then, there is the Croatian coast Dalmatia, the "warm and sunny Mediterranean coast" the author probably had in mind. This is a completely different region that has been classified as Csa using the Köppen classification, i.e. a region that sees dry, hot summers. Typical for Mediterranean climates, most precipitation occurs during the winter months, and rainfall can be substantial, which is why these areas still typically have an average precipitation between 750 mm (e.g. in Split) and 1000 mm (e.g. in Dubrovnik). While this may be wetter than you might think if you only know Dalmatia only from holiday photos or from "Game of Thrones", this is quite average as far as temperate climates go.
The European Environmental Agency provides a map that puts all this into a bigger picture. On this map of Europe, dark blue areas are those with higher annual precipitation. In central Europe, the Alps stand out as very wet regions (including the Slovenian alps), alongside the western coasts of Norway, England and northwestern Spain. But if you look carefully, Dalmatia does not belong to that area – the hue is much, much brighter.
So, the claim is not true. The Dalmatian coast is not one of the rainiest places in Europe. It is not "the rainiest place in South and East Europe as well as in the Balkans". What the article says about Slovenia is true, but the fact that "some parts of Slovenia can receive between 2000 and 2500 mm of rainfall per year" has nothing to do with the Mediterranean coast.