Yes, your understanding about the available number of PCIe devices is correct. However, it doesn't necessarily mean you can have as many NVIDIA GeForce RTX 2080 Ti GPUs.
Are # of PCIe lanes associated with the processor or the chipset in the motherboard?
Both. In modern computers, both CPU and motherboard (the South Bridge, SB) can have PCIe connectivity.
As you presented a Z370 MB, I'm taking Intel Core i7-8700K as example. According to Intel ARK, it has 16 PCIe lanes built in, which is very likely connected to the two PCIe 3.0 x 16 SafeSlots. This means you can have at most 2 PCIe devices that can communicate with CPU directly. The other 8 PCIe lanes (1x4 + 4x1) belong to the SB and are intended for peripherals like NVMe SSDs or network cards.
That doesn't mean you can't connect RTX 2080 Ti GPUs to the SB, however. Continue reading these answers to more questions.
How many PCIe lanes does a Nvidia 2080 Ti graphics card need?
This depends. If you're using those cards for machine learning or cryptocurrency mining, a single PCIe lane may suffice. But I haven't heard someone using fresh flagship card for those tasks, so I'm assuming you're using them for gaming.
For gaming, it's best for the RTX 2080 Ti card to run on full PCIe 3.0 x16 speed, as it will maximize the utilization of its outstanding performance. Running such a card with only 8 lines is likely limiting its potentials, as the data transfer rate is going to be the bottleneck when processing high-quality textures, or other I/O intensive tasks. If you lower some expectations, you may stand its performance under only PCIe 3.0 x8 speed.
How can these lanes be split? As in the above example board
If you don't get some extra pieces of hardware, you have only two options: 1x16 and 2x8 for the CPU lanes, and 1x4 for the SB lanes. This is because only the two biggest slots are connected to the CPU, so you can split the CPU lanes to at most 2 devices. The last big slot can also be used for a 3rd card if you like.
Without additional hardware, you can have at most 3 cards on that board, as there are only 3 "big slots" on the board (also take into account there's no RTX 2080 Ti GPU with single-slot thickness). But with some splitters you may be able to get more. See this thread for more information.
Best Answer
PCIe uses a point-to-point topology, so each lane expects one device on each end. If it wasn't, it'd be something like the original PCI - one issue with plain PCI is that when one device is talking on the bus, others have to wait.