Sometimes a teacher will simplify a process into "black and white" steps to make it easier for new people to learn the essential lesson. Hopefully the teacher introduces the shades of gray as students become more competent, but that doesn't always happen, depending on the way the information is presented (for example, a book or video vs a hands-on class) and the ability of the student and competence of the instructor.
IMO, the "close to the same temp" rule feels to me like a "black and white" rule created for first time soapers that never got expanded into the "shades of gray" version for more experienced soap makers.
It's not the temps of the lye solution or the fats that is important. It's the temp of the soap batter at the start of soap making. If the ingredient temps are within 10 degrees of the target temp, the soap batter temp will be close to that target temp too. That's the beginner rule -- easy, straightforward. A soaper can stick with this forever if that's what makes them comfortable.
I have never learned this rule, but if I had, I would have quickly found this rule to be limiting and tedious. If I want to shoot for a particular soap batter temp, I can balance warm fats with cool lye solution or vice versa to get close to my target. The temp difference between the fats and lye might be 30 or 40 degrees, rather than 10, but that makes no matter.
The shade of gray is there is more fat than lye in a batch of soap. The soap batter temp is always going to be more like the fat temp than the lye temp. If my fats are a bit warmer than my target temp, my lye will need to be quite a bit cooler so my batter hits the target temp. Or vice versa.
This is a lot of complexity to throw at a first time soap maker, but it's certainly something a person can learn after a batch or two.
