There is no problem with having multiple domain names registered to the same corporation. Problems arise when registering domain names to fictitious business names that are not properly registered with the Secretary of State and/or the county of residence.
This answer is provided for informational purposes only, does not create an attorney-client relationship, and is not to be relied upon as legal advice or applied to specific situations. Legal advice is provided only upon execution of a written retainer agreement and after a comprehensive consultation in which all relevant facts are discussed and reviewed.
There is a distinction between the TRADE NAME of a company, and the company's TRADEMARKS which are the symbols used in commerce to identify the source of particular goods or services. Trade names are not automatically trademarks.
Generally, a website name is not the assumed name of a company. For example, in the footer of your website you could say that the company name is ABC Corp., while the website itself is .
You may want to check to see if Minnesota's laws vary, but unless you are holding out the company and/or doing business under a fictitious tradename, you do not need to register trademarks as assumed names.
This answer does not constitute legal advice, and by answering this question we have not entered into an attorney-client relationship.
As you may read from the other answers, you do not need to register a domain name as an assumed name. Many companies have elected to do this for the economic process - in cases where payments are made, but you can imagine the management process for these many names and the attendant accounting that is needed.
Web sites are technically easier since one may only redirect traffic to a parent site or build content to suit.
An assumed name implies registration with the authorities and potentially managing the economic outcomes that arise from such transactions. If you are looking to build brand identity, then you should consult a professional to guide the process.
This is not a legal advice as I do not have an attorney-client privilege with you. You should retain a lawyer before acting on any generally available advice.