Does a new name for the internal HD make my Time Machine (and CCC) backups useless?

I have a replacement 1TB internal HD that I need to clone from my existing internal HD, using Carbon Copy Cloner. Initially I'll connect it in an external enclosure via USB, no questions on that.

I'd like to give the new HD itself a new name. Is there any way to have the new drive recognize all the corresponding data on my external drive Time Machine. Or will I have to start all over? I know that means I'd lose several months' worth of TM data.

As long as I'm asking about TM, does the same principle apply to the 1:1 clones made with CCC?

Thanks for any help.