I don’t understand what this has to do with anything. If the reason is “Sock Puppet”, you’re right it is because they believe you have at least one other account (same user - several accounts). The age of the accounts (used or unused since a long time) has nothing to do with this (unless you meant some old accounts were alt accounts from some people).
Yes, you may be right. It is totally possible you show up with the same IP as someone else, especially if you use a quite popular service (the same thing may happen to people logging at work from a big company, for example. It seems to happen often in certain communities attracting several people from some big companies during work).
Now, I don’t think the admin/moderators should ban solely because of an IP match, but maybe they do 
You should definitely see this with them in this case. Not much else to do. (Note: It shouldn’t be something automatic. This has been done manually by someone)