This verb reminds me very much of the South Park episode where all the girls form a club and rate the boys. When the girls hold a club meeting, the group members vote 'yes' on an issue by saying that 'it sparkles with me' instead of 'sure I like the idea'. See? In Irish it shines with the dog. In South Park, it sparkles with the girls. :D
You'd have to make some stretch to get "is enjoyment of it" from taitníonn sé - you don't have the verb tá for "is" and you don't have the noun taitneamh for "enjoyment".
If you're going to stretch things that far, you're probably better off stretching your understanding of "it shines" in English.
taitníonn sé lena madra.
"My", "your" (singular) and "his" cause lenition - mo mhadra, do mhadra, a mhadra
"Her" has no lenition - a madra
"Our", "your" (plural) and "their" cause eclipsis, but you can't eclipse m, so it's ár madra, bhur madra, a madra. For a word that starts with a letter that can be eclipsed, it would be ár gcapall, bhur gcapall, a gcapall.
For a noun that starts with a vowel, it's m'úll, d'úll, a úll
"her apple" is a húll, and ár n-úll, bhur n-úll, a n-úll for the plural pronouns.
In this sentence, the "possession" isn't in le, it's in the a of lena, - lena being "with her". Leave out the possessive pronoun "her", and you've still got "with" - le.
Taitníonn sé leis an madra - "The dog enjoys it"
Taitníonn sé le madra - "A dog enjoys it"
Taitníonn X le Y is just the construction used in Irish to say "Y likes X" (note the reversal).
Is maith le X Y can also be used for "X likes Y" (no reversal), but Is maith is used in a lot of different ways so it can be confusing too.