I had a couple Murray bikes when I was a kid. The Murray bikes and lawn/garden equipment were made by Murray Ohio Mfg. Co., which was based in Brentwood, TN on Hwy. US 31 (formerly Cleveland, OH), with manufacturing at the Lawrenceburg, TN (US 43) plant. I have seen both these facilities when traveling to MS. The name is now a trademark of Briggs & Stratton Co.
The Easy washing machines were made by Murray Corp. of America. This company was later known as Wallace-Murray, and also made Eljer plumbing fixtures among other items. They also made steel cabinets for Montgomery Ward at one time.
Both companies got their start in the automotive industry, supplying parts to auto makers. What connection they may have is unknown, but may be a result of someone starting a company, then selling part of it off in the early years. I couldn't verify it for these companies. So many of these old companies have some interconnection.