Typically new construction has nail fins.

They would of had to cut the stucco back to put them in with a fin.

If was new construction The Stucco would have applied afterword not leaving the gaps you have.

In new construction the nail fin would have been set in a wet sealant, then window flashing tape tied into the sheathing and weather barrier with proper pan flashing and drip cap. This step would be where the water protection comes. Afterwords the caulk can be optional. The foam would be merely to make it air tight and R-value

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@toddinmn Yes, they did cut the stucco back so that they can put them in with a fin, and they did the flashing. It was a new construction. The only thing is that the window contractor outsourced the stucco part to another stucco contractor, and that stucco guy somehow left the gaps that I have now (I guess he didn't do a good job). My question now is, should I just caulk around the gaps around the window to make sure that it's water-tight and air-tight, and in case somehow the water still get in, the only place that the water will come out is the two "weep holes?" Thoughts? Thanks again!

"Is it true that for "new construction window" installation, the edges usually would be left like that?"

No. Your windows are not properly installed. Period.

"I think for all vinyl windows (retrofit or "new construction"), the edges should be air-tight and water-tight, and the only place that the water can go out, is thru the weep holes."

Correct and you will have water issues again. The windows do not look properly flashed to me and caulking will not make up for that.

Please post your photos here in the comments not as links

Lot going on here. What does the contract specify? Stucco is a bit like opening up Pandora's box. No way to really guarantee the window won't leak unless you have some sort of rainscreening and through wall flashing.

@toddinmn The window is Milgard 8240T Tuscany Series V400 VNL VS.

Which photo you feel that they installed the window that was meant to have siding go behind the flange?

Here are some photos shows the flashing detail before the stucco was patched in:

For sure they didn't install the top drip cap flashing (metal) piece, but maybe that's optional? See: https://www.phoenixmetalsltd.com/how-to-install-drip-cap-flashing-for-windows

Another separate (but related) question. According to the window installer, he said that our old house wrap (or waterproofing building paper) is pretty old (91 years old to be exact, those brown-colored old paper), so he suggest that next year maybe we should consider re-stucco the entire wall. Any thoughts about this? Thanks again!

BTW, my favorite "how to properly install a window" youtube video is "This Old House" Tommy's demo: https://youtu.be/EOzkimq9ipo?si=CPpkKsM79MPEHzMF. 

The 8240T has a 1 3/8 set back from the nail fin, meaning the window will protrude to the exterior that amount.

It also has an integrated 3/4” J channel which is often done with vinyl siding so you dont have to add a j -channel.

The window that seems that would have been best in this application is the 8230T. This window has a 1” set back and a stucco key. This would have allowed for a drip cap as well.

" For sure they didn't install the top drip cap flashing (metal) piece, but maybe that's optional? "

No it is not "optional". Drip cap or some sort of head flashing is window install 101 and absolutely required.

The metal part is optional and not typically done with integrated J-channel. The head flashing is the most critical, then the bottom then sides. A window with intergrated J was not the right choice IMO. The finished result is flawed. Hard to say how well done the flashing was installed.

These are merely observations from pics and not being there.