Sonic the Hedgehog came out on June 23rd, 1991. Many people consider this day his "birth date." Since then, Sonic's birthday HAS been celebrated.
It's a boy! Sonic The Hedgehog - The one that started it all. Sonic the Hedgehog is technically Sonic's birth.
Happy 6th Birthday, Sonic! Sonic Jam - Sonic Jam could technically be called Sonic's 6th birthday. It pays a homage to Sonic's then-famous Sega Genesis games, and even features exclusive artwork. Though it isn't exactly a Sonic game, the Sonic World feature in it is quite a feat. Sonic Jam would be a predecessor to Sonic Adventure.
Happy 10th Birthday, Sonic! Sonic Adventure 2 - Sonic Adventure 2 is explicitly Sonic's 10th anniversary, even featuring a 10th anniversary set that was released on Sonic's birthday, but only in Japan. See more about it here.
Happy 15th Birthday, Sonic! Sonic The Hedgehog (2006) and Sonic: Genesis - Though both are lackluster games, they still give Sonic a... "happy" 15th birthday.
Happy 18th Birthday, Sonic! Sonic 4: Episode I - Sonic 4 was codenamed "Project Needlemouse," a direct homage to Sonic's prototype. It can also be noted that the game was originally going to come out during the summer (more specifically June, the month of Sonic's birth), but was pushed back to make way for further development.
Happy 20th Birthday, Sonic! Sonic Generations - Sonic Generations is the only Sonic game to explicitly feature Sonic's actual birthday.
Happy 23rd Birhtday, Sonic! Sonic 1 3D Remake - The most classic way to celebrate Sonic's birthday is with a remake or port of the one that started it all, but a certain previous attempt to do this brought disrespect to the tradition. The Sonic 1 3DS Remake allows you to go through the levels in 3D, nothing special.
The Next Birthdays
Sonic's 25th birthday is coming soon, and everyone would be especially happy considering Sonic's finally hung on for a quarter-century! It's quite obvious that there will definitely be some attempt to give Sonic his best birthday yet. I'd like to see what people think about this, but I personally think that Sonic's 25th birthday will be a special episode of Sonic Boom, the Sonic Archie Comics, or something else. However, I really do look forward to Sonic's 25th.
By the way, I'm also makin' a YouTube video for Sonic's 2014 birthday.