Due to this post I did ask if they were married at all, but it is only mentioned once in the movie that were husband and wife. After Jessie is rushed to the hospital one doctor ask the other if she has relatives with her, to which the other reply's "Her Husband". So yes they were married.
1. Defacto relationship. If you live with a partner for two years you are considered married in the eyes of Australian law.
2. Not wearing wedding rings. Maybe their religion forbids it. Maybe they decided not to. Max might choose not to as it can get snagged on car parts, and Jessie decided not to either.
There are probably many more storyline explanations for now rings- maybe they couldn't afford them, maybe Max is protecting Jessie by not advertising he has a family, etc. etc....
In reality it is surely a small detail the costume department missed. Nor did they ever imagine fans would someday be going through the film frame by frame to glean every nuance of the story.