Does yogurt maker keeps object at optimal temperature maybe around 40 centigrade, or current temperature of object?

There are a variety of ways to make yogurt. One that might be "easy" would to use a thermal cooker (giant thermos flask) if you happen to already have one, which would be under the "maintain temperature" (actually temperature will fall, but slowly) classification - but if you already have one, it's no extra cost or equipment needed.

Others have a thermostat and heater, but it's rarely an adjustable thermostat. I am currently using a dehydrator which is adjustable as my temperature control device; I have in the past used an oven with a standing pilot light (not adjustable, but a good temperature.)

Regardless of which type you use, you need to cool the milk (after scalding) to 35-40c before adding the yogurt culture (a small amount of any yogurt with live culture) or you will kill the culture - so you can't just dump 80C milk into "whatever" and expect yogurt, regardless of temperature controls or lack of temperature controls.

My research and personal testing W.R.T. the tradeoffs of time and temperature has lead me closer to 35C (for up to 24 hours) than 40C as a default setting. On the other hand, some go as high as 45C.

I built a yogurt incubator, so I know a bit about how they work.

Yogurt incubators without temperature controls will have a preset temperature, and they'll heat the contents to this point with a thermostat or a temperature controller such as a modified PID controller. (Engineering: normal PID controllers are not suitable for cooking, since the heat-up phase will create a large integral error term, which will result in compensatory overheating.*) For a consumer product, it wouldn't make sense to hold an object at its current temperature, because that would assume the user set the initial temperature correctly. It would be a food safety issue to let the user implicitly set the temperature, without giving clear controls.

Next, it's not possible to hold an object at its current temperature if you don't know what that temperature is. Yogurt incubators often (usually?) allow you to use your own container, and there's no good (inexpensive) way for the device to read the liquid's temperature without touching it. If it used a typical thermometer (whether it's a thermistor or thermocouple), the milk would cool down by the time the device managed to get an accurate reading. On the other hand, if you are setting the temperature (arbitrarily heating to 43°C, for instance), all you need to do is measure the temperature of the plastic housing or the air--eventually, the milk will match this temperature.

*Note: my yogurt incubator does use a PID controller, but it's my product rather than a consumer product, so it only needs a certain amount of idiot-proofing. If I don't put overly hot or cold milk into it, it won't overcompensate and overshoot the desired temperature.