That's not at all true. Spectre can most certainly access memory from other processes, including on AMD.
What they are referring to is Meltdown, which is specifically a privilege escalation exploit that allows a user process to access kernel memory from within it's own virtual memory space. Spectre, on the other hand, tricks another process to leak it's protected memory.
Even then, the Spectre paper specifically mentions how it may be possible to use it to access privileged memory by targeting an interrupt or syscall.
And AMD may very well turn out to be vulnerable to Meltdown too. While the researchers weren't able to get their PoC working on AMD CPUs, they did show that they *do* out of order execute instructions following an illegal memory access and discuss the problem may just be a matter of optimizing the side channel method they used.
Honestly I think AMD is being very dishonest in their announcement, beyond just the Meltdown handwaving. They claim the Spectre bounds check bypass has been fixed with software, but I haven't heard of a good software solution to this, much less have I seen an actual patch. Then they claim the Spectre branch target injection isn't an issue, but my understanding is this is just a matter of figuring out how to better mistrain AMDs branch prediction, as was done with Intel's.
These vulns are much more difficult to develop than your typical software vulns, and the researchers have barely even scratched the surface. There's sure to be much more to come and AMDs claims to be largely immune are horribly irresponsible. Until they disclose their actual reasoning behind their claims, I'm going to assume they're full of shit and just as vulnerable as everyone else.