As some people pointed out:
Number of polygons: If two games do the same thing, but one has more polygons (and assuming the artists are just as good), the other will have a dated look.
Art style: I personally loved the old 2D graphics. 2D games at 1999 managed to beat graphic-wise the 3D games of 2005 in terms of how good they look. Nowadays, the old 3D games look terrible, and 2D still just looks outdated.
Complexity of objects: If a battleship used to look toward the enemy, and suddenly a fiery hell will shoot out of it, we now expect to see the turrets move, target the enemy individually, and each one shoot.
Details: Using the battleship example again, when it being hit by the enemy fire- an old game will have the shots disappear, or have a general "battleship has been hit" effect. We now expect to see an explosion when the missile hit, and air flowing out from the breach, etc (or well, in a the near future).
In regards to FE, while it does not compete in art style with other games (it's quite unique, for good and bad), so we can't compare polygons, I'm not sure I can say that the art style look "current".
In terms of complexity, there's a dated look when people compare to other games such as total war (in tactical), or the world when compared to civ. Some of it comes from the whole customized units part, but it's easy to forget.
FE strongly fall on the Details part though. Just concerning the picture that came in the post: the water/beach looks like someone took a saw, cut a piece of land, and put it in some blue ooze. Inconsistent shadowing- looking at the units and the bare trees at the buttom, we expect that the general light's source comes from up and north. The city, forest and the treasure site do not have a shadow, though.
Beside that, the world looks as if it's built on squares. comparing the forest/plains border, it looks as if someone told the trees to stop.
The land generally keep being the same besides a few terraforms also bothers a bit. What, nobody ventures into the wild or works the fields? (as opposed to the upgraded terrain in civ)
While I didn't notice all of this, this is what came up to my mind when actually thinking about it. Part of it is choice, but part of it is what makes the game looks dated.