Not an answer, but I'd like to consider the question from another angle. It's pretty unlikely that a language would have separate everyday words for the physical notion of mass vs. weight. But, once its speakers start to learn modern physics, they obviously need different physics jargon for mass vs. weight. Then the question becomes: Is there any language whose everyday word for "mass/weight" is their physics jargon for "mass", instead of "weight"?
Korean has an analogous example: 힘(him) is an everyday word for "force/power/energy", used in many expressions like "apply a force" (e.g., push something), "is powerful" (e.g., of an engine), "conjure up energy" (e.g., to do homework), "exhausted/out of energy", and so on. However, when you do physics, you obviously can't use the same word for force/power/energy.
Hence, him was chosen to mean "force" in physics. For the physical notion of "power", Korean has a separate jargon, 일률(illyul), but it's such an obscure word that I've never seen it outside of physics textbook.
The end result is that, if you ask "Does Korean use 'force' instead of 'power' to describe how powerful a machine is?" then you could legitimately answer yes. However, it would be also somewhat misleading - it doesn't mean Korean speakers have a deep understanding of the underlying physical concepts and somehow find "acceleration multiplied by mass" (=force) a better match than "energy divided by time" (=power).
It just means that when the first Korean physics textbook was written, the authors decided the word him was a better match for "force" instead of "power", and then they obviously had to make up a different word for the physical concept of "power".