No, you don't need to.
a) Amazon Unlimited Kindle was created for those who liked the Kindle Owners' Lending Library service in Amazon Prime, but didn't like or want the rest of Amazon Prime's services.
For what it's worth, I gave Kindle Unlimited a try not long ago. While the idea is great, it isn't as good as I expected. For a comparison between those two services:
Kindle Owners' Lending Library (through Amazon Prime)
- allows you to borrow 1 eligible book or audiobook (only the ones with 'Kindle Owners' Lending Library' icon) any time.
- you can't borrow another book until the borrowed book is returned.
- you can have the borrowed book on your Kindle as long as you like. I once forgot to return mine until a year later.
Kindle Unlimited
- allows you to borrow up to 7 books and/or audiobooks (only the ones with 'Kindle Unlimited' icon) per month from Amazon.
- when you borrow those books, you can only have them for one month. Some can be rolled over to the next month, but some will be automatically returned. It depends on each book.
- browsing for eligible books is surprisingly tough. You can restrict search results to Kindle Unlimited eligible only, but it's erratic as not all are actually eligible. I found this so irritating that I ended the one-month trial.
b) you don't have to subscribe to either Kindle Unlimited and Amazon Prime to read a free book on your Kindle. You can download free books to your Kindle from Amazon (e.g.
Amazon Top 100 Free Books) or a legit site elsewhere, e.g.
http://kindleclassics.blogspot.co.uk or
this list of Kindle-friendly sites. It's a good idea to download a book management program like
Calibre to keep track of your free ebooks. It's free.
c) you can borrow ebooks and audiobooks from your local library to read on your Kindle. Most libraries in this country offer those you can borrow (often up to 8 ebooks/audiobooks any time) at no charge through OverDrive (
https://www.overdrive.com).
But yes, you don't have to subscribe to Amazon Prime or Unlimited Kindle to read a free book on your Kindle.
Frankly, using your local library's Overdrive service is much better than Amazon Prime and Kindle Unlimited. Some books do have a waiting list (each library have a limit on how many users can access each ebook), but there are other ebooks you could borrow from your library while you wait for your turn to read that book (it usually either alerts you that it's ready to be downloaded or it'd be automatically sent to you - this depends on each library).
Hope this helps.