We have a Sitecore 8.0 instance that we want to replace by a fresh new Sitecore 8.1 instance. So I installed a new 8.1 instance, installed the necessary modules and TDS packages and the site is running fine.
However we have lots of Analytics data in the old 8.0 instance that we don't want to loose. Is there a way to get that Analytics data into the new 8.1 instance?
Can I just add the old 8.0 analytics database to the ConnectionString and remove the entry of the new 8.1 one?
You should check the Sitecore 8.1 upgrade guide (available on dev.sitecore.net). It has a section on how to upgrade the data by rebuilding the reporting database (this process might take a while). This way you should be able to continue with you old data.
Attach your new site to (a copy of) the reporting database, perform the upgrade steps for that database as described in the upgrade guide (running sql script) and follow the steps for rebuilding the database.