Back before the creation of Sonic, SEGA didn't really think about having a mascot, nor did they consider having characters appear in multiple titles. They took the approach of using character ideas in a release, and then moving on to new ideas. When the company saw Nintendo's success with Mario, they decided they wanted to move ahead with the idea for a company mascot.
Naoto Oshima, the original character designer for Sonic, opened up about how the company came to settle on the ideas that would become Sonic and his friends/enemies.
“I planned a trip to New York while this discussion was going on internally. They said ‘We definitely want to see something like an old guy with a mustache, we also want to see something spiky, and we also want to see a dog-like character.’
The hedgehog was the most popular. People pointed at it and liked it. Second was Eggman. Third was the dog. This was kind of pleasantly surprising. I was asking myself ‘I wonder why it is?’ The conclusion to me ... was that by a lot of people choosing the hedgehog, it will transcend race, gender, different types of people. I reported it back to the company.”
As you can see, the rest is history. Well, except for the dog. I guess it just wasn't popular enough to move ahead with. I think it might be time for SEGA to resurrect that dog!