In the US, almost exclusively, we pronounce it as ' feb-yoo-wary' without the 'r.' I think in some British accents it's pronounced with the 'r' as in the transcript you're referring too...like 'feb-bru-ree.'

You're right, it's regional/dialect differences. Some (especially North America) might say it like you transcribed: ˈfɛbjʊəri while other (like parts of the UK) might say it more like ˈfɛbr(ər)i When looking for transcriptions, it is important to identify where it's from

I am British and people say both. Pronouncing the middle 'r' is actually considered correct so would probably be used more in professional situations. However, I do say the middle 'r' and it hardly makes any difference in practice.

In the United States, both pronunciations are common. The "roo" form tends to be heard when people are trying to be careful about pronunciation, in formal situations. I started to write my own explanation, but the usage note in the "American Heritage Dictionary" is perfect so I'll just quote what they say: "The preferred pronunciation among usage writers is [the "roo" form], but in actual usage [the "yoo" form] is far more common and so cannot be considered incorrect. The loss of the first r in this pronunciation can be accounted for by the phonological process known as dissimilation, whereby one of two similar or identical sounds in a word is changed or dropped so that a repetition of that sound is avoided. In the case of February, the loss of the first r was also helped along by the influence of January, which has only one r." As a child I certainly started out saying "Feb-yoo-wary" because I thought it ought to follow the same pattern as "January," and needed to be corrected. A similar situation exists with the word "library." In the US, most children, and some adults, pronounce it as "lie-berry." In this case, though, American Heritage does not include this pronunciation, while Merriam-Webster calls it "nonstandard."

In the British English you do pronounce a hard 'r' sound which would be like 'wr' sound so: Fe-wrar-i - this is how i learned it in school. Though Sasha, as long as you are understood by others, the way you pronounce is not an issue, because there are too many dialects of English. American, British, Irish, Scottish, Australian, and so forth.