Exactly. I was thinking it referred to a pretty big toy, like a playhouse, car, or something along those lines.

From what I understand, most nouns, except for ones that describe people in particular, are the same whether they're the singular or plural forms. The 들 particle only really works for 사람, or say, 의사, possibly 선생.

Apparently, the purpose of tongue twister sentences like this is to provide training in brain-tongue coordination to assist in learning to pronounce Korean words correctly?

The plurality is implied, so there is ambiguity, but languages vary like that.