"My brother-in-law loves food."
Translation:Svogeren min elsker mat.
The omission of possessives is mostly done with relatives you're expected to have only one of; mor, far, mamma, pappa, svigermor, svigerfar, mormor, morfar, farmor, farfar, bestemor, bestefar.
If you have only one aunt or uncle, it's fine to refer to them as just 'onkel' and 'tante', with no possessive, but if you have more than one then it's customary to add their name: onkel Per, tante Kari, etc. to avoid confusion. Still no possessive needed.
For siblings, brothers/sisters-in-law, and cousins possessives are used.