ProDOS is not firmware, as I understand it. It is disk-loaded.
Many of the Flash-based disk drives for C64 have a DOS shell that includes the CMD disk and file management functions (CMD made hard disks for C64 back in the day), and a bunch of other DOS enhancements built in. So, yeah, kinda; I think it's as close as one could get to being the same. C64 never had a disk-loaded DOS, as the DOS was built into the disk drives themselves (often enhanced by cartridges or a pluggable firmware called JiffyDOS from CMD), so there is not a direct parallel to ProDOS. But, certainly the enhancements from a bunch of places have become widely available to anyone still using the platform with any modern upgrades.
So, yes, you can work with directories and large collections of files, today, that could not be worked with on original hardware (and realistically there wasn't enough room on disks to need good file handling and directory support in the DOS back then). There's also a shell that makes common operations a lot nicer.
Edit: I think it might be worth explaining for folks who weren't around the first time that DOS stands for "disk operating system", and it was exactly that...it made disk drives work, and provided commands for working with files on disks. The OS itself, on both C64 and Apple, was mostly in the firmware, and was mostly just the BASIC programming language and some boot routines to setup the hardware for software to use. There was no protected mode, and no real notion of APIs, as provided by Operating Systems today. Many C64 programs would map over the built-in ROM, and remove what built-in functionality the OS did provide (like BASIC) in order to get more usable RAM. GEOS was perhaps the closest analog to a modern operating system for those old 8 bit systems; in that it had an API and did provide an abstraction of the hardware, to a small degree. Apple loaded more from disk than the C64 did, in the general case; making it more flexible, but also slower to start up.
Many of the Flash-based disk drives for C64 have a DOS shell that includes the CMD disk and file management functions (CMD made hard disks for C64 back in the day), and a bunch of other DOS enhancements built in. So, yeah, kinda; I think it's as close as one could get to being the same. C64 never had a disk-loaded DOS, as the DOS was built into the disk drives themselves (often enhanced by cartridges or a pluggable firmware called JiffyDOS from CMD), so there is not a direct parallel to ProDOS. But, certainly the enhancements from a bunch of places have become widely available to anyone still using the platform with any modern upgrades.
So, yes, you can work with directories and large collections of files, today, that could not be worked with on original hardware (and realistically there wasn't enough room on disks to need good file handling and directory support in the DOS back then). There's also a shell that makes common operations a lot nicer.
Edit: I think it might be worth explaining for folks who weren't around the first time that DOS stands for "disk operating system", and it was exactly that...it made disk drives work, and provided commands for working with files on disks. The OS itself, on both C64 and Apple, was mostly in the firmware, and was mostly just the BASIC programming language and some boot routines to setup the hardware for software to use. There was no protected mode, and no real notion of APIs, as provided by Operating Systems today. Many C64 programs would map over the built-in ROM, and remove what built-in functionality the OS did provide (like BASIC) in order to get more usable RAM. GEOS was perhaps the closest analog to a modern operating system for those old 8 bit systems; in that it had an API and did provide an abstraction of the hardware, to a small degree. Apple loaded more from disk than the C64 did, in the general case; making it more flexible, but also slower to start up.