In these phrases 'tóir' is 'popular' but in many comments I see people using 'keen'. Foclóir.ie has 'a bheith tóir air' for 'popular' and Teanglann defines 'tóir' as pursuit, but does have 'Tá tóir ar spórt acu, they are keen on sport'. Are these sentences in Duoling standard usage?
Note that "in these phrases", it is tóir ar, not just tóir that gives the meaning "popular". This has nothing to do with "Duoling standard usage", that's the way that the phrase is understood by Irish speakers generally.
Both "popular" and "keen" are adjectives, but they work in different ways - "they are keen on sport" = "sport is popular with them". "keen" is modifying "they" but "popular" is modifying "sport". As the Irish sentence uses a noun (tóir) rather than an adjective, the most appropriate adjective to use in English depends on who the subject of the sentence is.
It might help to think of tá tóir uirthi sa Ghearmáin as "she is sought after in Germany", where the link with "pursuit" is more explicit, but note that that would not be a good translation, because that's not how Irish speakers understand this construction, it's just a handy mnemonic to help you to understand how tóir ar ends up as "popular".