Can a private limited company have two managing directors? If not, what post/designation does the other person get for having an equal share in the business?

A company can appoint 2 or more directors, but there can only be one managing director. The other director can also be an executive director, but not the MD.

As far as the equal holding is concerned, it has no relation with the directorship. Holding shares gives you title of shareholder. So, the shareholders need to decide who will be responsible for what. If the shareholders want to give equal positioning, both can become executive directors of the company and no MD.

Managing Director: As per section 2(54) of Companies Act, 2013.

Managing director is director, by virtue of the articles of company, or arrangement with company or a resolution passed in general meeting or by its board of directors, is entrusted with substantial powers of management of affairs of the company and includes a director occupying the position of managing director, by whatever name called.

I hope this help you. If you still have any query, feel free to write in comments.