The short answer to your first question (in my opinion) is no, ASICs and FPGAs are not really equivalent (generally) in the context of how they are used in different applications. ASICs are baked-in circuits; once you make them, they are fixed. You can make the circuits and digital logic in them dynamic at runtime (for example, a CPU, or a more complex example, a domain-specific runtime accelerator like Sohu). An FPGA lets you reprogram the circuit inside it (via one level of extra abstraction, the logic and routing configuration inside it). In essence, an FPGA makes the ultimate tradeoff of being fully reconfigurable down to the "gate" level at the cost of other things like clock speed, area, transistor size, power, and so on. These tradeoffs may be so significant that it looks like GPUs and ASICs are worth it for deep learning inference rather than FPGA.

In general, you can implement deep learning accelerators on both FPGA and ASIC. Xilinx and now AMD have been slowly adding more and more stuff to their "FPGA chips," like an AI engine (vector processors and a network-on-chip), as well as high bandwidth memory, in addition to their configurable logic already there, to make it more viable as a possible solution for companies to deploy deep learning stuff to their devices or in applications where they want to integrate deep learning alongside regular FPGA processing stuff. I don't know how that shakes out in the industry, but I do know that lots of academics use FPGAs as a good platform for experimenting and prototyping accelerator architectures and such.