Is it necessary to use Honda transmission fluid in a 1993 Accord?

Nah, just use whatever fits the spec. I used Motorcraft (Ford brand) Dextron III in my 1991 Accord. No problems whatsoever. Feels maybe like it slips a bit more, not quite as hard of a shift.

I did read that Honda fluid is a slightly thinner fluid, so with the Dextron III you might get slightly harder shifts on a cold morning. Just make sure you don't cheap out and get crap fluid, pay the extra $2 and get stuff worth your while.

The above answers are poor advice. Use only Honda ATF Z1 or Honda DW-1 ATF. Notice they both said it did not shift perfectly. That is exactly the problem with using anything but Honda fluid. Honda even states in their owners manuals that you can use Dexron III in an emergency but flush it out as soon as possible. A new transmission will cost you around $3,500. Do you really want to take a chance using a fluid when Honda specifically states that you should only use genuine Honda fluid? There is really no debate about this. Use any other fluid at your own risk.