Just to be clear, I do understand that the units of the rate constant k is selected to make the equation dimensionally consistent. That is not what I am asking.
It should be the case that k 'means' something physically, which is implied by its dimensions/units. And I would also assume that what k 'means' should not be dependent on the order of the reaction, since I cannot see any obvious reason why it should. But we know that the overall order of a reaction changes the units of the rate constant, so I must be wrong.
My question is as stated: Why do the dimensions of the rate constant change depending on different reaction orders, and what does that physically mean?
As an extension to this, I would also like to understand how the same thing works for the equilibrium constant, where the stoichiometric coefficients of the species or the number of products/reactants change the units of the constant. Again, I cannot see the reason why these would change what the constant means dimensionally.
Sorry if this an obvious question. I am not a chemist.
Edit: I wanted to clear up a few things. First of all, as I understand I was expecting there to be a deeper meaning to the dimensions of the rate constant even though there isn't always an obvious one, since it is an empirical equation which merely simplifies a process rather than describing the fundamental workings of it. But I still want to understand intuitively (if there is indeed an intuitive reason) why the units are different for each overall order. As in the examples Maurice gives in his answer, a reaction of order zero produces dimensions of amount per time whereas a first order reaction produces [ratio] per time and so on for other real number orders. It just seemed strange to me that a constant in the equation, which seems to be doing the same job regardless of order, express something in completely different dimensions depending on order.
After reading your answers, this is what I thought: In a zeroth order reaction where the rate is constant, amount per time seems to be the only reasonable way to describe the rate since ratio per time inevitably produces a curve. Similarly for a first order reaction, amount per time wouldn't work because this value changes constantly. So, the constant is expressed in terms of the units which can describe the reaction given its nature. This was stated/suggested in some replies but thinking of it over an example is what cleared it up (if it is correct) and thank you for your responses. Perhaps this was obvious but a constant with varying units was not something I was familiar with.