CC photo from anniemack on Flickr.
The above photo is of Vin Scully, who has been the Dodgers' broadcaster since 1950. For many in Southern California, a love for the Dodgers and a love for Vinny go hand in hand. Fans across the nation feel much the same way about their own long-tenured baseball announcers (many of whom are now deceased), whether it be Harry Kalas in Philadelphia, Jack Buck in St. Louis, Harry Caray in Chicago, Jon Miller in San Francisco, and so forth.
But football teams don't have their own exclusive TV announcers. The economics of each sport dictates its broadcasting style, due to differences in national TV exposure and advertising.
For the majority of fans, baseball is a regional sport. Only occasionally do fans see national broadcasts of random matchups on ESPN, Fox, and other channels. Only two teams--the White Sox and the Cubs (and until recently, the Braves)--have anything resembling a regular national TV presence. Only recently could diehard fans follow out-of-market teams through premium services such as DirecTV or MLB.TV. But the local team can be seen day in and day out, so it makes sense for the club to have its own announcers to build a sense of consistency, warmth, and intimacy with the viewers.
Every Sunday during the NFL season, each game is seen in the local markets of the participating teams as well as other parts of the country, often nationwide. In both baseball and football, local broadcasts are blacked out whenever there's a national broadcast. This is because the league earns more money from national advertisers who can reach the most earnest viewers in the local markets, as well as the rest of the country, than each team could earn from local advertisers on its own broadcast. Because every game is seen by fans across the country, there's no room for each team to have its own TV broadcasters.
On the radio, teams in both sports have their own announcing crews. In the playoffs, both sports employ a single TV broadcast for each game. Advertising and national exposure are the rationale for both.