Linux is not Unix, literally. Both operating systems are similar in operation and application, but they aren't identical, even though many people use them interchangeably.
What Then is Unix?
Unix is an operating system that was first developed at AT&T's Bell Labs in 1969 by Ken Thompson. Unix was originally written in assembly language, but after Dennis Ritchie developed the C programming language in the early 1970s, Unix was rewritten in C. Apart from Ken Thompson, Unix's developing team included Richard Stallman and Dennis Ritchie.
One of the fundamental propositions of Unix is a multi-user multi-tasking operating system. Since computer systems were very bulky in the 1970s, it was difficult for many people to access them at their mounting locations. To provide more access to limited computing resources, there had to be a way to do many things (multitasking), and accommodate many people simultaneously, while keeping their access independent of each other (a multi-user approach).
With Unix, users could access the same computer system across different terminals, each user being independent of another.
Unix is proprietary software, and like all proprietary software, it isn't free. Free here doesn't refer only to cost ($, more on this below). Unix was portable across different computer systems because it was built in C. This was an important feature and one primary reason for its wide adoption. Unix developed quickly, especially because of its use in academia, and soon had various licensable variants and derivatives from different vendors, each offering unique features for specific applications. One such derivative is the Berkeley Software Distribution (BSD).
Linux Is Not Unix
Linux is a recursive acronym, it means Linux Is Not Unix (the Linux here also means Linux Is Not Unix, and so on). Many people use Linux and Unix interchangeably, and while they share many similarities, they aren't identical. Linux was built on the principle of free software. After the early stages of Unix's popularity, Richard Stallman started a project that he envisioned to be a free alternative to Unix.
Free in this sense refers not only to cost.
You may incur some costs in getting free software, but the fundamental principle behind free software is the freedom to use, modify, distribute, and personalize it as you want.
The beginning of what is now popularly known as Linux started in the late 1980s as the GNU project. GNU is a recursive acronym. It means GNU is Not Unix (literally, gnu is an actual word. It's the name of a large African antelope, and the head of this antelope is the logo of the GNU project). The goal of the GNU project was to build a free alternative to Unix. GNU built many free tools, but the project was not a complete operating system, it lacked a working kernel.
In the early 1990s, Linus Torvalds built a working kernel on top of earlier-built tools from the GNU project. He called the kernel Linux. GNU and Linux were two different projects, but their combination gave life to the free operating system now known as Linux. Hence, the operating system is more appropriately called GNU/Linux.
Free Software vs Proprietary Software Analogy
If you go to a quick-service restaurant to buy food, you may enjoy the food, but you most likely will not have the liberty to dictate how it should be prepared, or how it should taste. You may go to another QSR to get a different taste of the same food, but you still don't dictate how it should be prepared or how it should taste. This is like proprietary software (Unix, in this case).
In contrast, if you get a meal kit from a subscription service, you are free to enforce your preferences. The kit may come with recipes and suggestions, but you ultimately decide what to prepare, and how to prepare it. If you like spicy food, then you add as much spice as you want. You are under no restriction in choosing a recipe and don't have to get a license. This is like free software (Linux, in this case).