Dog and Person are structurally the same so you can assign a person to a dog and vice versa. But that's just how structural typing works and as a user of TypeScript I haven't run into a case where this'd be an issue.
I worked on a web based editor. A library would give us a range to highlight, in 1-based coordinates. The editor control was 0-based. As you can imagine it was easy to forget to translate back and forth in one path or another. In a strongly typed language I would simply define two Range types and the compiler would eliminate the mistake. I assumed Typescript could help me in the same way but it allowed the two types to be interchanged silently because they had the same structure. Perhaps I was holding it wrong?
Typescript has two hacks that help with mixing of similar data and introduce somewhat-nominal typing - branding and flavoring [1]. Also see smart constructors [2] for more functional approach.