Before I saw the comments/annotations:
GAME - Why I think it is.
1st video
Need for Speed Underground
Need for Speed III
Need for Speed Hot Pursuit 2
Test Drive Unlimited 1 - GPS, next one
Test Drive Unlimited 2
NFS Carbon
Burnout 1 or 2 - Never played the first two ones, but recognize the song
Colin McRae Rally 2.0 - Played on PS1
Midnight Club 3
OutRun 2006 Coast to Coast - I played this game for the first time THIS YEAR
Some anime ********. Don't even care, I hate anime. Something related to Initial D?
Need for Speed Most Wanted - obviously
Need for Speed Pro Street - played a lot, liked that you had to pay for damages
Crazy Taxi
Need for Speed Underground 2
DriveClub - By end I WAS thinking it, because of weather. But I only played it once in a friends house, so I'm not very familiar with it.
2nd video
Driver
Dirt 3 - Gymkhana, car and song
Burnout PARADISE... - (...City) Also the car
Need for Speed High Stakes - My favorite NFS on PS1, except for the songs (WHY? With every other NFS before it they got it right, Rock AND Techno, to suit more tastes)
Project Gotham Racing 4 - Song, car and weather (which is so heavy and complicated it can ONLY be done on next-gen, right FORZA?)
Juiced - Car
First I really didn't know (Nice song, though)
Burnout 3 - First Burnout I played
GRID - Car (song is from trailer bu I don't think is in game)
Test Drive Unlimited 1 - NOW I MEMBER! That song.
Cruisin' USA - Never played, but was expecting the song to say exactly the title at any time
Street Racing Syndicate
Gran Turismo 2
Gran Turismo 3
...WHAT? Song sounds like it's from Mario Kart or something
...Motorstorm? Never played it but now kinda want to
Midnight Club - I thought it was the first one though, didn't remember that song from the second one, which I played quite a bit
Enthusia - Cars remind of GT, song kinda too, but it's a different one. Never played this game, but heard about and wanted to.
GODDAMMIT AGAIN? - Japanese cars are nice, but why they like such weird electronic songs
Driv3r - Song, beach front, car not as much though
Burnout 2 - That one I'm sure about, due to car and song
3rd video
... - (It was Tokyo Xtreme Racer 3!?) Funnily enough, I actually played this one, just not to the end. Never heard this song in it, but there was one I actually liked, look up "Vampire Killer"
Forza 1 - Never played, but I know songs from it (WHY did Forza 4 made the same mistake as NFS High Stakes on the soundtrack???). Also IT SAYS SO ON THE CAR
Driver 2
Nope
Colin McRae Rally 4? - I played it on PS2
Nope, again
Really? - Song gives it away (DAYTONA USA)
Twisted Metal - Car + song, + crashing of course
Midnight Club 1 - Played it on GBA a bit, was awful, but had this song as well
ROAD & TRACK PRESENTS THE NEED FOR SPEED!!! - Song + car
Nope
Nop.. wait... - ...is it Auto Modellista? Yes. Livery (with an A) gives it away
Nope
*sigh* - The SONG SAYS IT (CRUISIN' WORLD)
Nope - (It was Stuntman, I never played the first one)
Forza 2
BEST STREET RACING GAME EVER!!! - Even on new-gen, NFS still can't have first person, REEALLLYYY!!??!?!???! Not to mention how that m*f*ker is always-online.
Oh isn't it awesome how in this game the car's mileage gauge ACTUALLY SHOWS HOW MUCH YOU DROVE THAT CAR, like some cars in GT6!
Don't know, is it that Test Drive Off Road on the PS2? - WRONG
Test Drive 4 - Cars remind me of classic Test Drives, which I barely played (only the 6th one I played a lot) - WRONG, it's 5, not 4
Nope
Blur - Racing at night, looks overall.. wait.. the f*ing License plate!
BONUS
The beat is sounding like that Peter Gunn theme... oh wait its Mich.. its the beginning of GTA Vice City!!
Nope
Need for Speed Underground 2 Customization shops
Gran Turismo 1 car dealerships - Though it did not have THAT Aston Martin, it's the second one that did
EDIT: Woo, this was a biggie.