Since when do rats attack people or a pose a threat in any way?

Yeah, they were obviously afraid that the rat would bite the baby. And when you think about it, rat bites can be awful even to an adult (they must really hurt, and we don't want to know what kind of bacteria it can spread). But when I last saw the movie, I had the same reaction as the OP, wondering if the rat really could be that dangerous. And in the abbreviated audiobook version of this story, that I grew up with, Tramp became a hero during a fire instead. But it seems like babies can die from rat bites. This is what David and Melissa from "The Disney Odyssey" have to say about this:

Unfortunately, even to this day you hear rare reports of babies or children being bitten by wild rats in bed, resulting in injury or even death, thus this concept alone is frightening. Consequently the rat is a very believable threat, and does not need any more of a character development than what it gets. The scene between Tramp and the rat is terrifying to watch.

First of all, the movie is of course told through the dogs' perspectives: In the scene w/the rat, Lady means to protect the Baby, and Tramp is to help her do just that. He may have a cynical attitude toward babies, but rats--they're a whole different story. And he's probably dealt with them before, living on the streets.
But is the rat a direct threat to the Baby? No--more of a potential threat than anything. There is no proof whether or not it was lurking off screen just waiting for Jim Dear and Darling to have a Baby in the hopes that one day they'd leave it unattended with an incompetent caretaker who would leave the window open so it to climb in and bite the infant to death--No.
More likely it was probably trying to get out of the rain, like any sensible animal. During the fight with Tramp, a horrified Lady (who has broken from her chain and has just entered the nursery), sees that the rat has climbed up on the crib. Is the rat actually thinking, "Now I'm gonna bite that baby to pieces and there ain't nothin' you can do about it"? Not likely. It was probably attracted by the warmth of the blankets, the smells (soiled diapers, spit-up, etc., which could be enticing to a rat). However, if it had gotten into the crib and found the Baby, its first instinct would likely have been, "What's this strange, hairless creature doing in my new sleeping quarters?" Then seeing as this "intruder" has no teeth or claws to defend itself with, the rat would've attacked it. Just my two pennies' worth.