Let $A$ and $B$ be topological spaces (or smooth manifolds with possibly empty boundary if it matters, I'm not sure either way).
Let $X$ be a fixed ambient topological space (or ambient manifold if it matters), i.e. $A, B \subset X$ (and $A, B$ have the subset topology, if that matters). Then I know that:
$A,B$ isotopic in $X$ $\implies$ $A,B$ homeomorphic $\implies$ $A,B$ homotopically equivalent.
Also, in case it matters, when I say "isotopy" I think that I mean "ambient isotopy", i.e. the sense used in knot theory, see for example Grumpy Parsnip's comment here: Isotopy and homeomorphism. Also maybe relevant: Isotopy and Homotopy.
My question:
Now assume that the ambient space is not fixed, i.e. we want to consider all possible ambient spaces for $A,B$ (is this the same as all spaces in which $A$ and $B$ can be embedded?). Then do we have
$A,B$ isotopic in some ambient space $\iff$ $A,B$ homeomorphic? (or $A,B$ diffeomorphic?)
Obviously one direction is true, but I am not sure about the other. I was watching Tadashi Todieka's lectures on algebraic/differential topology on YouTube and sort of got the impression that the other direction might be true, but he never said so either way.