Well the phrase "trophy wife" doesn't exist on this web page and I would say that the expression itself is not sexist (because you could also have a "trophy husband" and everyone would understand what it means), but of course the practice's prevalence suggests a cultural sexism where the expression doesn't.

The term "sugar baby" comes idiomatically from the opposite of the terms "sugar daddy, sugar mama" as the person who is the target of the affections of the latter. Obviously, "sugar baby" is not a sexist expression because it is not gendered. As for what it means, to be clear: Alice is Bob's sugar-baby (conversely, Bob is Alice's sugar-daddy) if Bob is an affluent older person who is both dating the younger Alice and is gifting her with financially-expensive gifts. The gifts could be as immaterial as "I have this car I never drive, you can use it as if it were your own" or as material as "I will pay off your student loan debts." Usually, if Alice has her own professional career and is providing well for herself, then we'd not call it a sugar-baby relationship even if the gifts are expensive and one-way. Furthermore if she is taking care of the household and/or kids, that would tend to mollify the judgment. The essence of the judgment is the belief that if the sugar-parent were not providing the gifts to the sugar-baby then the relationship would not persist; the very terms themselves suggest that the relationship is too uncomfortably invoking a parent-child dynamic in a romantic context.

A stay-at-home wife is generally not a "sugar baby" since they are married and not dating; you could potentially extend the term to cover married couples but even so, if there are no expensive gifts or they're of comparable ages you wouldn't normally say that he's her sugar-daddy.