Aunt Sarah, especially to people who aren't dog lovers. It's possible she's just prejudiced against dogs and doesn't like them. It's also possible she didn't want to allow any animals near the baby, since she did leave her cats downstairs.
The alligator in the zoo. Trying to eat Lady or trying to help remove the muzzle - and simply not realising his mouth was too big.
The rat gets this too, to a much lesser extent. Was it truly trying to kill the baby, or was it only focused on using the house as a shelter?
Anvilicious: Only irresponsible troublemakers dream of a life that doesn't involve staying where you are, raising a family and doing what your owner tells you.
Catharsis Factor: After an entire film of Aunt Sarah misunderstanding and vilifying Lady's behaviour, when Jim and Elizabeth finally return and release their dog, she once again freaks out about the dog being on the loose and being a threat to the baby. Jim tells her she's speaking nonsense and deduces (correctly) that Lady is trying to tell them something important (that Tramp saved the baby's life by killing the rat).
Designated Villain: The dog catcher is perhaps the least evil of the Big Bad Ensemble; his job is to find dogs and put them in his pound to await pickup, but the film generally portrays him as antagonistic for this, particularly when he agrees to have Tramp put to sleep. Also, though she may be particularly harsh towards Lady and the Tramp, Aunt Sarah does what she does out of concern for others, particularly for her cats (who get Lady muzzled with a Wounded Gazelle Gambit) and for Darling's child. Plus, she hates runaways (and also, it would seem, strays), to the point of chaining Lady to the backyard doghouse to teach her a lesson, and around the time she arranges for Tramp's euthanasia, she's only aware of two things as far as the surrounding circumstances were concerned: one, that the child's life was in danger, and two, that two dogs, one of them a stray, were at the scene.
Ethnic Scrappy: The infamous Siamese Cat duo. On top of being, of course, villains, their Ethnic Scrappy status manifests itself in their buck teeth and the annoyingly terrible grammar during their song.
Fair for Its Day: The ethnic stereotypes personified in animals, the cats in particular.
Narm: When the rat is bitten by Tramp the first time, it squeaks like a chew toy. (It squeaks normally every other time, not even close to narmy.)
Family-Unfriendly Aesop: Aunt Sarah insisting only cats should be left with babies. Leave it to Lady to prove she can. Now here's some real-talk: If you pay any remote attention to the news, no baby or young child should ever be left unattended around animals. NOT EVER. Doesn't matter how well trained you think they are, or if they're not a dangerous breed, animals will get jealous of the attention the newborn gets and will maul them.