The answer to your question depends on what you mean by the term access.
If you are talking about read-only access to the file system, then it is theoretically possible that a program may be able to offer this, so long as the permissions of each file allow reading by the role "other." However, as @Izzy pointed out in response to your self-answer, this is not the case on a normal Android system.
In order to have full access (read/write/execute) to all files on the system, you must either be their owner or the super user, which means you must be root (which implies that you must be using a rooted device).
Why this is the case
Android uses a Linux file system in which all inodes (files/folders) have multiple permission bits (called "modes"), so different levels of access permissions can be assigned to each file by role (owner, group, other). For example, a file might have the following permissions:
-rw-r--r-- root root some_file.conf (This is the same as mode 0644 in octal representation)
Given these permissions, any user on the system may view (read) the file, but only root may alter it (write). No app running in the normal userspace will be able to alter this example file. It can only be altered by root (the super user).
These are just simple facts of how a Unix file system works (you can read more here), and no app can violate these basic rules.
What this means
If the app you mention in your answer is actually able to gain full access to all files on the system, then it is somehow gaining super user access on an unrooted system. When an app gains super user access to a device that has not been rooted, this is called a privilege escalation exploit, and it represents a serious security concern. Such an app needs to be reported to Google so that it can be removed from the Play store, and so that the vulnerability it is leveraging can be identified and patched.
If you simply mean that the app allows read-only access, then I would argue that is not true for standard Android systems (as pointed out by @Izzy), and your question is ultimately misleading. Read-only access is not what most people mean when they say they want to "access" their Android file system.