Does he mean to say that my engine doesn't disengage completely when the clutch is released, if that's the case then my clutch would wear out. Am I correct in thinking so?
You should be pushing the clutch pedal to start the engine no matter what time of year it is. There is a clutch safety switch which needs to be engaged in order for your ignition to work when you turn the key. It can only be engaged when you press down on the clutch pedal. This is to prevent you from trying to start the car while it's in gear. Just a safety feature they installed in cars a long time ago.
Starting the engine in Neutral (manual gearbox) without depressing the clutch means the starter will have to get the engine parts moving, but also the main (input) shaft on the gearbox. Cold temperatures affect not only the oil in the engine, but also the lubricant inside the gear box. Thus the starter motor will have to contend with the slightly thicker oil within the gearbox at lower temperatures.
As you drive, the gearbox oil heats up progressively due to friction between moving parts and also the oil itself getting into motion. As it heats up, it also thins out, giving better lubrication and making it easier for the driver to change gears (the gears enter better). Transmission oil also has the double viscosity grading system, such as "EP 75W/90": 75 when cold, 90 when warmed up.
When both the engine and the gearbox oils are at working temperature, the engine may be started in Neutral without depressing the clutch. However, as pointed out by @Paulster2, best practice is still to depress the clutch when starting.
I personally always start the car with my foot on the clutch pedal as a matter of good practice.
If you start the car this way then you put very slightly less strain on the starter motor as it's not having to spin the entire clutch assembly or any shafts within the gearbox.
When I learnt to drive in the 1960s, British engines were not as powerful and batteries were nowhere near as good. I was taught to start in neutral with the clutch out and later, as a driving instructor, I passed this on. I don't know about now, but back then, depressing the clutch put MORE strain on the motor. This was easily established by listening to the motor, or watching the tachometer. If I had the car idling in neutral, and I depressed the clutch, the engine revs would go DOWN. Of course, living in coastal suburban Australia, I did not have to deal with the kind of cold temperatures that would be experienced in the USA or Europe.
There are two sides to this topic: 1) As you depress the clutch the input shaft is disconnected from the crank -> Less strain on the starter and the battery/electrical system.
This is all nice and dandy, HOWEVER: 2) By pressing down the clutch pedal you put an axial load on the crankshaft. This will lead to increased wear on the thrust bearing (The one crank bearing that also limits/sets the axial play of the crankshaft) as there is no or not enough oil pressure yet.
Personally, I use the clutch out neutral start. However, I never drive my cars for such short times where an increased strain on the battery can't be recharged. If the starter should at some point fail from the added stress, it is easier and cheaper to replace.