Well.....looks like you have been cheated IF you paid for drip edge.
There should be drip edge on all roof installations...that ain't it.
Drip edge goes up the roof under the shingles and down the fascia (where the gutters go)and into the gutter. If it does not go into the gutter it drains behind them...leaves nice streaks.
The metal you are looking at under your shingles is to hold the metal(aluminum) fascia piece and lock it in so you do not have to see any face nails or wavy metal. This is common for a siding contractor to do..not a roofer. A good roofer knows what drip is, a good one installs the 6 inch up the roof style.
This could be one of those communication problems we talk about with contractors, I would get them back to find out what you paid for first.
Call your building inspector and find out what is going on.
Then....it,s off to
If you don't use drip edge you will end up with issues...especially in your area of the country.