The short answer to the question of whether an e-book needs an ISBN is that there is no requirement at all for any book to have an ISBN number as such, regardless of whether it is a printed book or an e-book. If you don't want a book you author to have an ISBN, then no one is going to make you – but you'll very likely be on your own. You will have great difficulty in getting anybody in the book trade to take your book on, even if you are self-publishing, without an ISBN. It is one thing to publish a book, but the very practical problem of getting your book noticed by anybody, be they people that will be your readers or those that can get your book in front of the readers, is the real problem to be solved.
The ISBN numbering system exists for a very practical reason. It is there to help a book get through the whole of the book publishing system. For a printed book that means from author to customer, by way of publisher, printer, distribution, warehousing, marketing, stock control, retailer to point-of-sale. For e-books, that publishing system may be a little different, but the same practical reasons apply. So the answer to whether you need an ISBN for an e-book is yes you probably do. But there a couple of very popular special cases, namely Apple iTunes iBookstore and Amazon Kindle Direct Publishing that you might need to consider.
If you want to publish a book for sale through the Apple iTunes iBookstore it depends on whether your book is free or for sale. If you're offering it for free, then it does not need an ISBN. If you are selling your book, then the iTunes iBookstore requires that your book has an ISBN.
If you want to publish a book for sale only through Amazon Kindle Direct Publishing, then you do not need an ISBN. Amazon does recognise ISBNs but also have their own numbering system to keep track of products that they sell.