‘This language doesn't have types’ and ‘This language only has one type’ are English sentences that communicate the same underlying concept: a typeless language doesn't have a way to distinguish categories of values from each other, either statically (before running the program) or dynamically (while running the program).
It's not entirely clear to me, but you or that person might be conflating types (which are categories of values) with data structures (which are arrangements of data in memory that can store values). Basically all programming languages have data structures, because all programming languages need to represent values in some way if they want to have any practical use. How types interact with the data structures supported by the language will vary.
(A type system usually refers to the rules for how types are defined, checked, and related to each other. A language with types has a type system, even if it's an extremely basic one defining a fixed set of types which are not related in any way to each other. But more powerful type systems allow for complex types and complex rules for judging whether a program uses values of those types correctly. As a matter of terminology, whether we say a language ‘has types’ or ‘has a type system’ depends only on where we want to put emphasis: on the types themselves, or on the ways they interact and get checked.)