Regarding FPGAs, I always took this to mean just the general logic (the look-up tables and flip-flops) and interconnect in the FPGA (i.e. the "sea of gates"); That is to say not the hard IP blocks such as DSP blocks, ADCs, PLLs, or transceivers. It kind of looks like woven fabric with all the uniform square-grid criss crossing.
I've not really heard this used for processors but since you're talking about a soft process on an FPGA, I assume it means the same thing.
The quote "...and that will erase data within the processors fabric" lacks a little more context than I would like and seems a little ambiguous because I would normally take "clearing the fabric" as literally wiping the FPGA clean and the soft processor along with it. In that context it doesn't make sense to say "clear the data within the process fabric". If I took that phrase literally, I would assume it is talking about something similar to a hard reset on a hard MCU, where the data within the processor is cleared without wiping out the processor itself from the FPGA (something that is not possible with a hard processor anyways).