Other languages that I know would just use the simple present - j'apprends l'irlandais; ich lerne Irisch; ik leer Iers. It's quite difficult to think of a situation where you would say "I learn Irish" in English, or foghlaimím Gaeilge in Irish. (Or dysgaf Gwyddeleg in Welsh?)
There is a Dutch form which is almost identical in wording to the Irish version - ik ben iers aan het leren - but this is closer in meaning to the French je suis en train d'apprendre l'irlandais, roughly "I am in the middle of learning Irish", so is correct in fewer circumstances. It's also rather colloquial.
It's striking that the languages of the archipelago have developed such similar formations for this set of meanings. A (rather minimal) amount of googling indicates that some linguists think this is just coincidence. I am not convinced. I am reminded of the way that Albanian, Macedonian, Bulgarian and Romanian all put the definite article as a suffix to the noun, although they come from three different language groups.
Just a thought.