In the book "An introduction to manifolds" by Tu, a topological manifold is defined to be a topological space $M$ that is Hausdorff, second countable and locally Euclidean.
Does this allow things like the disjoint union of a plane and a line? Then we have a component which is locally Euclidean of dimension $1$ and one of dimension $2$?
On
Yes, it does. In the remark right after Definition 5.2, Tu states "Of course, if a topological manifold has several connected components, it is possible for each component to have a different dimension." The disjoint union of a plane and a line is a valid example of such a space. Note that the dimension will be constant on each connected component, so nothing more egregious than this example can happen.
Tu allows manifolds having connected components of different dimensions. He explicitly says it in this post. Usually people talk about a space being "locally $\Bbb R^n$" or "locally Euclidean of dimension $n$" as opposed to just "locally Euclidean", as he does. But it is not hard to show that for each $n \geq 0$, the set $$\{ x \in M \mid x \mbox{ has an open neighborhood homeomorphic to }\Bbb R^n \}$$is both open and closed in $M$. So this means that the dimension is well defined on each connected component of $M$.