One of the critical components of seeing a life as an economist is the ability to see non- economic events into an economic context. I think that this might be arising from a social economist point of view, as one who views contingency as reflective of economic theory. Karl Marx and other economic thinkers in this vein did this in quite a persuasive and strong manner. The ability to connect economic ideas to social elements such as arrangement of power structures or prevailing attitudes might bring one closer to thinking like an economist. It seems to me that the ability to apply economic ideas to social domains help to underscore the importance that economic theory plays a vital role in how we perceive the world and our place within it.
All economists have to think the same way to some extent. This doesn't mean that they have to come to the same conclusions but rather that they have to go about things more or less the same way.
Essentially, the idea is that economics is supposed to be a science and so "the economic way of thinking" is supposed to be a systematic approach to questions. It's supposed to be one where you form hypotheses and create models to try to explain reality. Like one textbook I've taught from says "Economists use the same scientific method used by other disciplines such as biology, chemistry and physics."
So it's not that economists have to agree but that they're supposed to think scientifically.
All economists use scientific methods, and all economists view all social or other events in the economics context. But these characteristics don't really help us to distinguish thinking of economists from those of others. Scientific methods is used by people practicing hundreds different modern disciplines and profession like physician's, historians, astronomers, criminologists, psychologists, engineers, anthropologists and statisticians. Also, person in every profession or field tends to connect different things, which are apparently not a part of his or her discipline, to his subject. Remember the famous case of Newton thinking up of the idea of gravity base on observation of an apple falling from the tree, or of Archimedes jumping out of his bath tub shouting 'eureka' when the buoyancy felt by him in the tub water, led him to think up of a scientific method to determine the purity of gold in crow of King Hiero.
When we say thinking like of economist, it should point to ways of thinking exclusive to economist. No other profession should follow the same set of attributes, though some attributes may be common with other professions but not most or all. This exclusive component of thinking of economists is economics itself. Samuelson and Nordhaus define economics as:
The study of how societies use scarce resources to produce valuable commodities and distribute them among different people.
In pursuing this type of study the economists ave developed many different methods and an theories of economics, which all economists learn. It is these common objective of study combined with grounding in the common economics methods and theories that give economists their typical way of thinking.
Samuelson P.A. and Nordhaus W.D. 2005, Economics, Eighteenth Edition., Tata McGraw-Hill, New Delhi.