While the claim is certainly false for thermoplastic polymers like polyethylene, it's far more likely to be true for thermoset polymers such as polyurethanes, epoxies or phenolics such as Bakelite (which, being the first widely used synthetic plastic, may well be the original source of this claim).
Specifically, a thermoplastic polymer, by definition, consists of moderately large (that is, large on an atomic scale, but still microscopically small from a human perspective) chains or branched "trees" of monomer units, each (mostly) linked to two other neighboring units in the chain.
(In polyethylene, for example, these chains generally have molecular weights ranging from a few hundred to a few million AMU, with an average typically somewhere around 10,000 AMU (≈ 1.6 × 10-23 kilograms) depending on the manufacturing process. For comparison, a single ethylene monomer weighs about 28 AMU, so a 10,000 AMU polyethylene molecule has about 350 ethylene units, or about 700 carbon atoms.)
These long chains stick to each other by intermolecular forces (mainly van der Waals forces) and also simply due to the chains getting tangled with each other (steric effects, in chemistry jargon) rather like a bowl of sticky noodles. When heated and/or mechanically stretched, however, the chains can slip past each other, allowing the material to be reshaped (= thermoplasticity).
In a thermoset polymer like Bakelite, however, the monomer units can (and typically do) link up with more than two other monomers. Thus, instead of forming chains, they form a complex three-dimensional network structure. As long as the average number of links per monomer exceeds a critical percolation threshold (which varies depending on the structure of the material, but is typically not far above two), most of the monomers will link up together into a single giant network that spans the whole object.
Once the monomers have joined together into such a network, they can no longer be separated without destroying the material. Thus, thermoset polymers cannot be re-molded after curing, and they tend to be relatively hard and brittle compared to thermoplastics (although exceptions exist).
(Another way to make a thermoset polymer is to start with a suitable thermoplastic polymer and somehow, e.g. by rapid heating and/or chemical treatment, create additional cross-links between the chains than join them up into a single network. Examples of such thermoset polymers include cross-linked polyethylene and vulcanized rubber. An advantage of such polymers is that they can be shaped like thermoplastics before curing them into their final shape and consistency; the degree of cross-linking can also be controlled to adjust their flexibility and elasticity properties.)
Thus, if your ice cream tub happens to be made of, say, polyurethane instead of polyethylene, it probably does consist mostly of a single giant molecule. While there might be some small isolated clusters that are disconnected from the main network, those will generally make up a relatively insignificant fraction of the material.
Ps. The best sources for this stuff are probably textbooks on polymer chemistry like  or introductory online sources like  or ; it's considered basic enough that you won't really find it discussed in primary sources like scientific papers these days, except maybe as background material.
In a lot of sources, you'll find somewhat weaselly wording like "the entire molecular network can be considered to be a single molecule" (, emphasis mine). One reason for such wording is that, while it's easy enough to conclusively demonstrate that a thermoplastic like polyethylene does not consist of a single molecule just by melting it, there's no way to definitively prove that any given piece of thermoset polymer really consists of a single network, and not, say, two or three or 10,000 interpenetrating giant networks.
While a single network comprising most of the material is indeed expected on theoretical grounds, provided that the amount of cross-linking is sufficiently high, and while it is consistent with observed evidence, we don't really have any way to directly show that, say, this atom and that atom at the opposite ends of the object are really connected by a chain of covalent bonds.
Another reason for such weasel wording is simply that many chemists tend to feel vaguely uncomfortable using the word "molecule" for things so huge that you can see them with the unaided eye, instead preferring to call such materials e.g. "network covalent solids". Such macroscopic chunks of matter simply don't have much in common with the kind of small groups of atoms (from a few AMU up to say, a few thousand AMU; maybe a few million for polymer chemists and biochemists) that most chemists normally refer to when they say "molecule", even if they technically do fall under the same definition.
- : Peacock, Andrew. Handbook of Polyethylene: Structures: Properties, and Applications. CRC Press (2000). pp. 6–10. (via Google Books)
- : Tuttle, Mark E., Structural Analysis of Polymeric Composite Materials, Second Edition. CRC Press (2012). pp. 8–11. (via Google Books)
- : Woodland Plastics Corporation. "Understanding Thermoset Plastics". Web page.
- : Cramb, Alan; Salvador, Paul. "Experiment #2: Understanding Polymeric Materials, their Structure and Properties". Online course material from Materials in Engineering: 27-100, Spring Semester 2000, Carnegie Mellon University.