No, what's being described is a CRM (customer relations management system). A CRM is a customer account management system that has awkward ways of doing other things; a CMS is a publication workflow system that has awkward ways of doing other things.
The problem with CRMs is that they tend to fall into one of three categories: "enterprise", which usually means complex, inflexible¹ systems that lock you into their structure (and which usually run on Java); "cargo cult", which are poor copies of enterprise systems, missing most of the useful features because it was essential to mimic the more pain-in-the-butt stuff that any good ground-up rewrite would have seen as a problem in the original (but would require too much end-user training to fix in an upgrade), but have the advantage of running on low-cost PHP hosting; and "toy" (or, in start-up speak, "minimum viable product"), which sort of cover some of the basics badly, need to be vastly extended to be useful, and generally have a lot of outdated copy-pasta under the hood.
You know your business and its requirements. The actual log-in is sort of a "done deal" in PHP right now; the language includes functions specifically for hashing, verifying and re-hashing passwords (using bcrypt with good defaults and a cryptographically random salt). Best practices for things like login tokens ("remember me") and password resets are easy to find these days thanks to things like Stack Overflow and the Information Security Stack Exchange. You know what customer details you do and don't need, and whether old transactions should be stored in document style or relationally updated (should the old or new address appear on an invoice from three years ago, etc.). Development can be time-consuming, but realistically it's no more time-consuming to build from the ground up than it is to take a generic system and customize it to your needs.
____________
¹ "Inflexible" might be somewhat unfair here. You can make substantial changes to the system(s), but even the smallest changes (other than skinning) are usually Projects with a capital P.