Well,
@thomas suggested a rinko-bukuro (dedicated bike bag, usually for taking a bike on trains). Did you have one with you? (Expecting a taxi to have something on hand is like expecting the trains to supply cyclists with bike bags.)
Also, bike bags come in two designs--one type you take the front wheel off, the other type you take both wheels off, in order to make the overall package smaller (besides being fully covered).
For the second type of bag (both wheels off), I think most road bikes could fit in the back seat of a taxi, but I doubt the one-wheel-off type would. Also, the trunk/boot is not an option for LP gas taxis--the tank is there (big) and there's very little room.
A guerilla bike bag option is to stop at a home center or 100 yen shop and buy the largest available trash bags and a roll of wide tape. With a few bags and the tape you can cover everything (again with the front wheel off).
If your bike has fenders and/or a front basket or back rack (most 'mama-chari'), none of this would work.