Sure, except what about political slogans? Supporting a candidate you like by making your own versions of posters? Oh, wait, that's what Shepard Fairey got in (copyright) trouble for:
http://en.wikipedia.org/wiki/Barack_Obama_%22Hope%22_posterBut it's the control over derivative works (http://en.wikipedia.org/wiki/Derivative_work) that copyright gives you that's more obviously suppressive. All of the uses of Mona Lisa would be derivative, if Da Vinci's estate still owned the "intellectual property". All of the uses of Grant Wood's "American Gothic" would be derivative.
Yes, not all "derivative works" are worth it, but neither are most "original" works. 99% of everything is crap.
Copyright law is also used to suppress anything that the Rightsholder doesn't like: http://arstechnica.com/tech-policy/2013/03/gopro-can-fall-fr..., http://www.dailydot.com/business/dentist-bad-yelp-reviews-co... and many others.
This may indeed be an abuse of copyright law (copyfraud) but the laws are apparently intentionally written to allow this sort of thing. The practice is to not prosecute improper DMCA takedown notices, for example: http://www.techdirt.com/blog/?tag=copyfraud
So, yes, copyright per se doesn't tend to stifle free speech, but the concept of "intellectual property" gets used to get laws like DMCA passed, which are then promptly abused to censor people and their ideas. Moreover, the practice of lengthening copyright terms towards infinity keeps ideas out of the public domain, causing derivative works to not appear.
One more thing: Mickey Mouse Meets the Air Pirates. http://en.wikipedia.org/wiki/Air_Pirates If that ain't censorship, I don't know what is.