... the ones I just got, the casings are made by Michelin.
Technically, the entire tire is "made by Michelin," since B.F. Goodrich is simply a Michelin brand name. To know which plant produced the tire, you need to look at the DOT code. Plants formerly owned by B.F. Goodrich (before the Michelin take-over) now produce tires branded both Michelin and B.F. Goodrich, and existing Michelin plants worldwide also turn out tires under the Goodrich brand.
As with many multi-national corporate buyouts, Michelin have chosen to keep the Goodrich corporate assets, such as HQ, R&D labs, etc., intact and operating under the Goodrich brand and moniker, because (1) consumer perception plays a big role in purchase behavior (many people who have a "buy American" mentality will buy a Goodrich tire but would never buy a "French" tire) and (2) it is easier to sell the brand and/or the corporate assets to another player later if you keep as much separate as possible.
Sometimes companies don't grok this immediately, and pay dearly later. When British Petroleum (BP) bought out Standard Oil (Amoco, aka Standard Oil of Indiana), they immediately re-branded all Amoco and Standard stations as BP stations, and thus immediately lost market share to other brands perceived to be "U.S." brands (such as Exxon and Chevron). They corrected for it, only after the damage was done, by hanging signs in former Amoco stations under the BP brand saying "Amoco Fuels."
Never underestimate the market power of a well-established brand name.
In any case, the distinction now is purely artificial -- both Michelin and B.F. Goodrich tires are "made by" Michelin.