I would say that the first Pokémon games for the original Game Boy was as good of a swan song that four shades of green system could ever hope for.
In Europe, Red, Blue and Yellow was the last three games released for the original Game Boy and Yellow was the last game released for it in North America.

The Yellow version with its Anime tie-in at the biggest media franchise cradle is my favorite swan song for a gaming system.

Underreated game.Thought I might be the only person who remembered it.   

Super Mario RPG and Harvest Moon were strong finishers for the SNES.

Twilight Princess for GameCube

Persona 4 and Sakura Wars for PS2

Panzer Dragoon Saga for the Saturn

OG Xbox: Halo 2. This one might be a stretch because it came out almost exactly 1 year before the Xbox 360 but this represented the zenith of the original Xbox for me, and I still played this game long after the 360 came out

Xbox 360: Gears 3. This was the last 7th gen Xbox exclusive of any consequence that launched and it ended the original gears trilogy on a high note. Unfortunately, this next several years for XGS would be a dark age

Xbox One: Gears 5. This is the best gears of war game, and I’m sad it didn’t get more attention because it was attached to such a failure of a console.

PS2: MGS 3. This one was difficult because this gen ended on such a high note. But this might be a best metal gear game.

PS3: The objective answer would be the last of us, although I really do not like what direction Sony took their entire first party with the success of this title, which is middlebrow high budget cinematic “prestige” experiences with unremarkable/creatively safe gameplay.

PS4: Death Stranding. Completely and utterly unique game, and major respect to Kojima for taking this risk.

A standout example for me is the aptly titled The Last Story.

Releasing when the Wii was on its last legs, TLS combines an awesome action/RPG combat system with loveable characters, a varied and briskly paced adventure, and a sweeping Nubuo Uematsu soundtrack, coalescing to deliver a game one of the most accessible and entertaining games of its kind from that generation.

It's truly a shame that it largely flew under the radar; it's a wonderful game that sent the Wii off in style.

Breath of the Wild, Donkey Kong Country 3, Kirby's Dreamland 3, and Twilight Princess are up there for me as well.

PS1: Final Fantasy IX. Nothing else comes even close, since it is in my opinion the best game of all time.
PS2: Persona 4. Released years after the next generation of consoles was already out, especially in Europe where it came out in 2009. FF XII would be the other option.
PS3: The Last of Us.
NES: Kirby's Adventure
SNES: Chrono Trigger

Some of mine have been mentioned already, so I'll go with Metroid Samus Returns for the 3DS. While it didn't sell great, it marked the start of what has become a renaissance for 3d Metroid after over a decade of nothing and reinvigorated the franchise which had been dormant for 7 years since Other M. It looks great in 3D and proved the 3DS wasn't dead yet even though the Switch had already launched.

Aside from obvious choices, I would say "Rockman & Forte" for SNES. It came out in 1998.

For me, Perfect Dark on the Nintendo 64.
Reflections, impressive lighting and texture work really showcased what the Nintendo 64 could do, even if the framerate was less than ideal.

Another really significant one for me is Project Zero 2: Wii Edition.

Arriving in mid 2012, when the Wii was essentially finished, it was marketed as a port of a decade old PS2 game but is actually something far more fascinating; a full on remake with redesigned gameplay and presentation and an expanded story, that is honestly one of the best old school horror games I've ever played.

It's atmosphere and scares are absolutely phenomenal, and its finely tuned pacing, nerve-wracking gameplay, and moody presentation are all excellent. It's an absolute crime it wasn't released in North America and received little attention, as it's a worthy if unconventional swan song for the little console that could.