I wish the Democrats would give similar weight to the ACLU or similar organizations. Maybe this is why people say organizing Democrats is like herding cats.
Democrats: 30-16 in favor
Republicans: 51-2 in favor (1 not voting)
The DMCA was a Republican initiative legislatively, but passed on an unopposed voice vote and was of course signed by Clinton. Cosponsored by 7 Republicans and 3 Democrats; drafted by the office of Howard Coble (R-NC), who also pushed it through the committee he chaired.
If you want to assign blame by overall party control, both laws, along with the Sonny Bono Copyright Extension Act, were passed by a split government, with Republicans controlling both houses of Congress, and Democrats controlling the Presidency. So I guess they're "bipartisan achievements", for which we can thank the 1996-1998 "dream team" of Bill Clinton, Newt Gingrich, and Trent Lott, who set an inspiring example in pushing through so much major legislation rather than letting themselves be mired in partisan gridlock.
This shift is a very positive development, it means the bill is becoming impossible to pass in a Republican House (and where GOP Rep. Lamar Smith has discredited himself by leading fellow Republicans into the SOPA morass).
The RIAA types are even more welcome in the Republican Party traditionally, due in part to Nashville as you mentioned, and in part due to the significant influence of first Sonny Bono and now his widow Mary in Congress.
If this is a regression to the mean, when was the mean ever in effect? In the 1980s, Ronald Reagan maintained strong ties with Hollywood (even with people like Lew Wasserman who were active in the Democratic Party) and promoted their interests; in the 1990s, the GOP led the charge for the DMCA and Sonny Bono Copyright Extension Act. Arguably there was strong hostility in the 1950s-70s, when Joseph McCarthy thought Hollywood was full of Communists, and Nixon complained about it being controlled by Jewish Democrats, but that was a while ago, and for reasons (paranoia about Jews and Communists) that hopefully aren't going to recur.