No, we most certainly do mean anonymity is a basic part of digital freedom. The only way you could consistently argue that it's not necessary is to advocate prohibiting anyone, including individuals, treating you differently for anything you've said. Which is clearly impossible.
Your framing is really just right in line with the bog standard authoritarian pattern of redefining "true" freedom as something that can be comfortably tolerated and/or further chipped away at down the line.
Of course what you say (and do) affects how others perceive you. Just as Uncle Jim drunkenly announcing over Xmas dinner how that Hitler guy had a good plan will affect how everyone sees him in the morning, your twitter feed will have an effect on how your neighbours talk to you.
That's a good thing 99% of the time. Social strictures have been how humans managed society for thousands of years. It can lead to terrible things of course - but we like to think we have built systems of justice and governance to overcome them.
But suddenly making everyone invisible and able to speak and act without consequences seems a bad idea in the real world - i am not sure why it's a good idea online.
Well yes of course, your first paragraph was exactly my subpoint - declaring otherwise would be impossible.
From there it follows that it is indeed a proper freedom to be able to communicate while limiting the scope of who is privy to that communication, including limiting what can be attributed to some singular inescapable "identity". Saying that preventing this is "a good thing 99% of the time" is a baseless assertion. While this may appear similar to how social strictures have been managed for thousands of years - that's thousands of years of oppression of individuality and persecution for deviating from the herd. The West has overtly bucked that vein of collectivism, which I'd call progress.
Furthermore what you're invoking as some open-and-shut traditional state of affairs was anything but. Witness the concept of hearsay - where party B claims that party A has said something, ostensibly so that others will judge party A for it, but modulo B's reputation. Digital communications render hearsay moot - the default result of party B being able to prove to themselves that party A said something, is that party B can also prove to anyone that party A said that thing. Hence the need for the cryptographic property of repudiation, as the physics of digital information leaves no ambiguous middle ground.
Your framing is really just right in line with the bog standard authoritarian pattern of redefining "true" freedom as something that can be comfortably tolerated and/or further chipped away at down the line.