There are now five spaceships parked at the ISS

Yes it has, on at least one occasion in 2014, but I believe it may have happened even more than that.

The two Soyuz spacecraft are used to carry the crew to and from the station. While docked, they aren't used for much besides making sure they're ready in case the ISS has to be evacuated.

The two Progress spacecraft are resupply ships. They're launched with 2,000+ kg of supplies which is offloaded, and then are refilled with trash and other materials that will be burned up in the atmosphere after the Progresses leave. The Progress are also capable of performing boosts to raise the station's orbit to counteract atmospheric drag.

And finally, the Dragon is also a resupply spacecraft. Similar to the Progress ships it is loaded with 2,000+ kg of supplies which is unloaded on the station, but instead of reloading trash, Dragons are loaded with experiments and other items that are destined to be returned to Earth intact for study (Dragon being capable of surviving reentry and landing back on Earth).

Wear and tear over time increases the odds of a failure. A structural failure on a space station is, in technical terms, a "very bad thing." Remember the last days of Mir, when it was constantly catching on fire and leaking air everywhere?

Why can't they just pull a Ship of Thesus and just keep building new modules and decommissioning old ones until nothing of the original station remains?

Because most of the station's modules weren't designed to do that. It's far cheaper to make installation permanent than it is to design hatches that can be (edit: reliably) exposed to space again after 15 years of being attached to each other. The electronics of a lot of the station are outdated, shielding techniques and materials science for the outside shells have improved significantly as a result of station research, and they only have one robotic arm to disassemble the station while they had two to build it.

New modules, like the proposed private module that will replace the BEAM after the test is over and the Russian lab Nauka launching next year, will be designed to be able to be removed from the station because they'll have plenty of life left when the rest expires. They'll be incorporated into new Russian and private space stations that will replace ISS. But those stations will be custom-fit with modern tech and layouts suited for their new roles, re-building the ISS won't be necessary when we're done with it.

NASA's Orion is nearing completion, but it will likely never dock at the ISS; it's meant for deep space missions.

NASA itself isn't developing any craft for the ISS. They've contracted SpaceX, Boeing, Orbital, and Sierra Nevada to do that.

This whole NASA this-and-that thing is really getting old. NASA was never in the business of building anything, and they only had design roles here and there, but not continuously. They are mostly mission design and program management folks, not spaceship makers. The whole "none from NASA" could be even said about Apollo. They didn't build the damn things, for crying out loud, and they subcontracted most of their design, too. NASA's role mostly is, and has been, of administrating the programs and designing and running the missions, not of designing or building entire spacecraft/rockets. Saturn V wasn't a "NASA" rocket. NASA was, and still is, mostly a customer for hardware designed and built by others. They do a lot of work in providing very detailed specs of what hardware exactly they want, and they are the users of said hardware.

As far as similarities go, Dragon is as much a "NASA" ship as the Lunar Lander was, except that for Dragon the SpaceX does mission control, too.