You should check with your CPA, but generally, if you operate 2 businesses through 2 different entities (e.g., LLC, partnership, corporation, etc.), each entity should have its own EIN. In fact, you probably would, for liability reasons and others, have a separate LLC or corporation for each business. If you operate as a sole proprietor, you have unlimited personal liability for business debts.
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Yes, but I would strongly suggest that you form an LLC (or maybe a corporation) for each of the businesses, and each company have its own EIN.
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Yes, though the better approach for legal and tax reasons almost always is to form two separate companies, with each company owning a separate business. There are many potential advantages, including:
(1) You may desire to have the companies taxed differently (i.e., C corporation, S corporation, partnership, or sole proprietorship) to arrange your affairs to have less exposure to tax liability;
(2) You can make certain that any liability that arises with respect to one business doesn't attach to the other business; and
(3) It can make it easier to separate accounting records if you ever desire to market one of the businesses for sale.
I recommend you spend some "quality time" with a good tax accountant and a good business lawyer. They should be able to help you know the best way to structure this within about an hour or so.
Also, regardless how the tax accountant believes you should be taxed, remember that an LLC is usually (but not always) the best form of business entity for small businesses and that an LLC can elect to be taxed in any of the manners set forth above.
Of course, I'm not your lawyer, and this isn't tax or legal advice. The sole purpose of this is to help you have a meaningful conversation with (and perhaps choose) your professional advisors.