After doing some reading, it looks like this is a common problem and there currently isn't a way to merge the two accounts. The most official Microsoft response I found is on the Azure Feedback Forums. Scroll down and see the comment made by Ariel Gordon (posted just over a month ago):
Thanks for the question and feedback.
Our team owns the Microsoft account and Azure AD sign-in/sign-up experiences. We know that some experiences are confusing for some of you and we’re working hard to simplify them.
More specifically:
1) We know a number of users have multiple accounts with Microsoft, some they created themselves and others they got from their work or school. Today, using multiple accounts in Office 365, Azure.com or VisualStudio.com requires you sign out of one account and sign in to another. To address this we’re building the ability to be signed in with more than one account at the same time, in the same browser. This should start showing up on Microsoft web properties later this year.
2) There’s a small number of Microsoft business services that only support “Live ID” accounts, and not organizational accounts that are used for other business services like Office 365. Examples include MSDN and Volume Licensing. We’re working with these teams to add support for organizational accounts. This will allow everyone who already has an account they use with Office 365 to use the same account to sign in to these services.
3) Some users have two Microsoft-powered accounts with the same sign in string: one they got from their organization (when the organization uses Azure AD account) and one they created themselves (for example to access MSDN or VL). For these users, the sign-in experience to Office and Azure can be confusing. If you’re in this situation, the simplest solution is to rename the account you created yourself. You can use another consumer email address or get a new @outlook.com address. And later his year, you’ll also be able to use your phone number as a sign in string. Longer term, we’re considering different options to remove this overlap.
Keep the great feedback coming and please let us know if you finds other sources of account confusion. We’ll continue to monitor this thread as we refine our plans.
Ariel
Also, see this blog post for a good breakdown on the difference between a "Microsoft Account" and an "O365 Account".