So what I have done now is changed the email address of my user account at both WordPress and Gravatar?
Automattic owns both Gravatar and WordPress.com. A WordPress.com account is a Gravatar account.
What I am trying to ask is: whether of or not I have to have the same email accounts for my user account and blog?
No you do not need two but if you wish you can enter two different email addresses.
You have two separate email address settings at WordPress.com.
Blog notifications for the admin (like comment moderation, Likes, new subscriptions, etc) are sent to the email address at Settings -> General in the Dashboard.
Personal notifications (like comments on your post, subscription emails, and upgrade renewals) are sent to the email address at Users -> Personal Settings in the Dashboard.
When changing email addresses try changing it at http://wordpress.com/#!/settings/ first.
If that doesn't work, try Users -> Personal Settings in your blog's Dashboard.
Issues Changing your E-mail Address > http://en.support.wordpress.com/email-address/#issues-changing-your-e-mail-address
Note: After the change you will receive an confirmation email. Please be sure to check your spam/junk mail filter in your email client, if you cannot locate it.