Jeff Conine 1998 game-used jersey
The Monarchs wore a dizzying array of uniform styles during their existence, with little carryover from year to year, or even similarity between home and away uniforms some years. This gives the Royals a lot of options to work through when choosing throwback unis each year. The Royals said they based the 1998 jersey on a 1949 Monarchs style. Here's the 1949 team:
Doesn't look like an exact match. The jersey the Royals wore looks like the one in this shot of Buck O'Neil, but I don't know what year this photo was taken:
It's possible that Buck is wearing the 1949 road grays there, and the Royals turned it into a white for their game. I haven't managed to find another picture of a Monarchs jersey that looks just like this one. It's a pretty common photo of Buck, so I wouldn't be surprised if it was the basis for the 1998 jersey.
The Tigers came dressed as the 1935 Detroit Cubs, an obscure team that operated independently for one year only. The jersey number on a cub on the sleeve is a great little detail: