Can a computer use shift registers as memory?
Yes, in fact the Kenbak-1 (considered to be the world's first personal computer, years before the Apple //) used serial memory.  It was introduced in 1971 as a teaching machine.

It didn't use a microprocessor since the first one (Intel's 4004), had not been released yet when the Kenbak-1 was designed.  Instead the CPU was made up of discrete TTL chips.

The Kenbak-1 had only 256 bytes of memory, implemented using two 1024-bit MOS shift registers.  So it was rather slow, averaging only about 1000 instructions per second despite it's 1 MHz clock.

The instruction set was similar to the Motorola 6800, which would not appear until several years later.  It had two 8-bit accumulators, A and B, and a 8-bit index register, X.  The memory addressing modes were immediate (constant), memory, indirect, indexed, and indirect-indexed which provided more capability than most microprocessors introduced later in the decade.