It's a bronze bearing so anything from SAE30 and up should be fine.
SAE30 or similar oil is good if you can press it into the pores of the sintered bearing
material but that is a process that most people cannot do at the kitchen table
so I normally recommend a not-too-thick-grease. That's far easier to handle without
making a mess, it stays where it is put and it works fine.
Yes, they are all a bit wobbly, even more so if dry. It should not wobble around when rotating
by itself or when driven by the motor but it will almost always wobble a bit if
the platter is pressed with your fingers.
Martin