There's some good advice in there, given with a horribly wrong intention.
The good advice is to repeat the manager's argument, the way you understood it. However, if you do it to "loosen them up" and make them "more willing to work with you", then you're doing it for the wrong reason.
The right reason is to verify whether you understood them correctly. Maybe they meant to say something else. Maybe the did something else and you misunderstood. Maybe they said exactly what they meant and you understood it perfectly, but there's more to it. For example, they might have a perfectly practical non-technical reason for their argument.
Here's an example from my own experience: at one point, a manager argued that we should do something that would risk defeating the very purpose of a rather big and complex software we've developed, by allowing a group of future users to circumvent certain of its restrictions that are meant to make sure their stuff can be used on more than one target platform. I pointed out this risk and argued vehemently against it. The manager explained that the cost of migrating their existing work to our software, while maintaining those restrictions, would drive the cost of the whole project to 150% (or higher) of what the project would cost if we allowed users to do what he proposed. That certainly complicated the whole equation, even though his argument was still "technologically horrible".
TL;DR: When non-technical managers disagree with you, it might be because they have a need to assert their power and be respected by their subordinates, but that's a conclusion you should draw only if you've made a reasonable attempt to discard the alternatives. If you do happen to have a manager that behaves that way consistently, then maybe you should consider changing the manager and/or the company, instead of wasting your time and energy by trying to manipulate them.
Agreed. If you're using a tactic to get your way and with the assumption that you are the expert (and the other person is wrong) then you are missing out. In my experience, the best solutions came after those discussions where you get the other party to explain their motivations. The result was always better than I could have come up with myself.
When in a position of authority, I had the opposite problem: I often couldn't get people to disagree with me. This made it impossible to get to that discussion. (Maybe my ideas really were perfect, but I doubt it.)
The good advice is to repeat the manager's argument, the way you understood it. However, if you do it to "loosen them up" and make them "more willing to work with you", then you're doing it for the wrong reason.
The right reason is to verify whether you understood them correctly. Maybe they meant to say something else. Maybe the did something else and you misunderstood. Maybe they said exactly what they meant and you understood it perfectly, but there's more to it. For example, they might have a perfectly practical non-technical reason for their argument.
Here's an example from my own experience: at one point, a manager argued that we should do something that would risk defeating the very purpose of a rather big and complex software we've developed, by allowing a group of future users to circumvent certain of its restrictions that are meant to make sure their stuff can be used on more than one target platform. I pointed out this risk and argued vehemently against it. The manager explained that the cost of migrating their existing work to our software, while maintaining those restrictions, would drive the cost of the whole project to 150% (or higher) of what the project would cost if we allowed users to do what he proposed. That certainly complicated the whole equation, even though his argument was still "technologically horrible".
TL;DR: When non-technical managers disagree with you, it might be because they have a need to assert their power and be respected by their subordinates, but that's a conclusion you should draw only if you've made a reasonable attempt to discard the alternatives. If you do happen to have a manager that behaves that way consistently, then maybe you should consider changing the manager and/or the company, instead of wasting your time and energy by trying to manipulate them.