I'm a native English speaker. In my opinion "the dog owner" is grammatically correct but semantically different from "the dog's owner"
"the dog owner" is the owner of A dog or SOME dogs (indefinite)
"the dog's owner" is the owner of THE dog (definite, talking about a specific, singular dog)
Since the Swedish sentence is talking about a specific dog, then only "the dog's owner" is a correct translation.
For whatever reason, I am having difficulty memorizing verbs that begin with för-. It just occurred to me that perhaps these verbs are compound verbs or verbs created by verb-ifing (I know this is not a word) other words. Can someone please comment on this thought? Thank you.
Yes :) It's a compound verb I believe. It has been borrowed from Low German 'vor-' but ultimately it seems to come from Proto-Germanic for 'before' or 'in front of'. If you search for a Swedish verb which begins with 'för-' on wiktionary you can see the word etymology e.g. https://en.wiktionary.org/wiki/försvinna
There is also a page with Swedish words which use 'för-' as a prefix - https://en.wiktionary.org/wiki/Category:Swedish_words_prefixed_with_för-
Thank you, loukaki! I had not found this wiki page before. This helps a lot.