Collars of a manifold

Question

• How is the collar of a oriented smooth manifold $M$ with boundary defined? Is it simply a topological space $C$ such that $C= \partial M \times [a,b)$ with product topology and $a <b$, as this answer somehow states?

• Two oriented manifolds with boundary can be glued together along their common boundary (see "Introduction to Topological Manifolds" by John Lee for details). My lecture notes state that, if you want to glue two smooth oriented manifolds with boundary together and obtain a smooth manifold, "choices like collars are necessary." How so?