Your hunch was right. It’s typically some kind of shipping crate or (wooden) container, better known as a Liftvan in German (you might also see it written as two words, Lift Van).

In your context, Lifts are the containers that would house the personal belongings of Jewish emigrants for overseas shipment:
"Anyone planning to move out of Germany had to file a detailed inventory of everything they owned before they could get a permit to leave. The normal method of shipping was by having a moving company pack the entire household into an enormous crate that would then be transferred by crane to a ship and transported by sea. When the crate reached the United States, the household goods would be transferred to a truck and delivered to the new residence."
https://hoebers.wordpress.com/2015/03/25/an-inventory-of-a-l...

A lot of these containers, though, never reached their destination:
"Das Hab und Gut, verpackt in Container-großen sogenannten Liftvans, sollte den jüdischen Emigranten mit Speditionen über die Umschlagplätze in Bremen und Hamburg nachgeschickt werden. Doch tausende Umzugskisten erreichten ihre ursprünglichen Besitzer nie. Mit Kriegsbeginn 1939 liefen die Frachtschiffe nicht mehr aus. Die Kisten stapelten sich in den Häfen - und die Gestapo ließ sie beschlagnahmen."
https://www.ndr.de/geschichte/chronologie/NS-Raubzug-Wie-Naz...

This is why there’s now a whole database people can use to look for their (family’s) missing shipping crates:
https://lostlift.dsm.museum/

The term still exists, but I believe today’s liftvans usually aren’t as big as they used to be (if you haven’t already, take a look at the third image you see at the NDR link; they really were enormous!):
"Nicht für jeden Überseeumzug benötigt man einen großen Container bzw. ein Vielfaches dieser Größen; manchmal sind kleinere Transportbehältnisse ausreichend. Dann kommen die sogenannten Lift Vans für kleinere Volumen in Betracht. Lift Vans sind aus Holz, können individuell gefertigt werden und heißen auch Crate oder Überseekiste. Sie kommen immer dann zum Zuge, wenn Ihr Hausrat nicht mehr als 15 Kubikmeter umfasst. Mehrere Lift Vans werden in einen größeren Container zusammengestellt."
https://delta-umzuege.de/internationale-umzuegeuebersee-umzu...

Two US links:
"Liftvans – Better for smaller shipments (generally under 850 cubic feet)
Containers – Better for larger shipments

A liftvan is a plywood crate built specifically for international shipments. A standard liftvan is approximately 200 cubic feet (87” high x 87”x long x 48” wide) though liftvans can be made to fit any sized shipment."
https://www.rainieros.com/moving-guide/before-the-move/liftv...

"Liftvans are wooden crates sometimes referred to as overseas crates. These containers can hold a little more than 1000 lbs. of household goods. There is a wide variety of liftvan container sizes within the industry, however, the standard crate or liftvan will be 87” long, 45” wide and 87” high for a total volume of 197 cubic feet, which at an average density of 6 lbs. per cubic foot should hold roughly 1200 lbs."
https://www.nationalforwarding.com/assets/1/14/January_2014....

Cf. discussion.