"The code" allows wet vents but not through horizontal piping. And it can be any nearby fixtures on the same floor of a certain size, the fixture group requirement was removed. Of course, this depends on which code we are talking about, I'm mainly familiar with the IPC formerly UPC. Horizontal wet vents could still be allowed by other codes.
Your bathroom was probably constructed when that practice was acceptable. Doesn't mean you can do it now. In any case, wet venting is allowed for low flow fixtures draining into the sizable vent of high flow fixtures. You proposal for wet venting the washer through the floor drain's vent has the concept reversed, so it fails on principle. There would be a very good chance the washer flow would suction out the floor drain trap. It's hard enough to maintain a proper floor drain trap seal as it is.
The floor drain trap will need it's own vent, though once you are above floor level with the vent, you may make horizontal runs and tie into other vent pipes provided the sizing is appropriate. Some codes allow a mechanical venting device, which obviates the need to run the vent outside somehow.