BOTH are 'I've got flu' and 'I've got the flu' are right.
In British English, 'I've got flu' is correct. Influenza is a recognised illness, so it has no article. Just as you might say 'I've got bronchitis' or 'I've got pneumonia', you say 'I've got flu'.
American English uses an article, which is an older, more informal and rather more vernacular ( non- medical) expression.
We tend to use 'the' with old-fashioned, informal and even slang terms, such as 'I've got the runs' (diarrhoea) or 'He's got the clap' (VD) . Sorry to lower the tone here, but those are the best examples and most common ways that we use 'the' in such phrases.
I'm not sure why American English puts flu into the same 'grammatical category' as 'I've got the runs' ( ie informal/vernacular), while British English puts it in with named medical conditions 'I've got bronchitis'. That's just the way it is.
It is clear why you were confused, though. English courses for Russian speakers are based on British English, which is why you learnt the British form. Most English you come across on the internet is American English, which is why you encountered the American form there.
Also, as a general rule, British English speakers are far more aware of American forms than Americans are aware of British forms. This explains why Abigail told you that the American form is 'the correct way' and why Jerry said that the version without the article 'sounds very odd'.
Both sound fine to British and Australian people, such as Peachey and myself.