ClientCare should know my company name

I'd say you're both right - ClientCare should totally know more about you, but there are consultants who are supported users for bunches of companies. Seems to me if you're only a supported user for one company, it shouldn't be difficult to figure out the details.

Over the years on the consulting side, I got into the habit of always including:

Company Name, Report Suite, Username

with every first email to ClientCare@adobe.

Especially since you're a supported user!

I'm guessing there are some consulting companies that deal with multiple clients at multiple organizations, so that's why they ask. But if your email address is the supported user for only a single organization, they really shouldn't need to.