Is the resulting function of plotting two variables imply that all other factors are being held constant? Say, for instance, I plot two relationships (separately):
(a) Share of income spent on food vs. household income; and
(b) No. of houses in a unit vs. population of a city
Then, for both (a) and (b), I separately run a run a regression of the y-axis variable on the x-axis variable and plot these functions on each respective graph.
Can one say that the resulting regression functions in (a) and (b) holds all other factors constant; even though the function I am plotting isn't controlling for any variable?
For example, in the case of (a) can one say that the graph shows the relationship between share of income spent on food and household income assuming and/or holding constant other factors that could impact the relationship between these two variables, such as demographic compositions of households, etc.?
 A: No, plotting the two variables against each other does not constitute controlling for all other variables.  Doing this with two different sets of variables also has no effect on that fact.  
What you are doing is ignoring all other variables.  To better understand the distinction, it may help to read my answer to: Is there a difference between 'controlling for' and 'ignoring' other variables in multiple regression?  That post is more focused on regression models, whereas your case is about plotting, but the point is ultimately the same.  
If you want a plot of a relationship between two variables that holds all else constant, you will need a properly specified model.  Ideally, you would have experimental-ish data where you could guarantee that all other variables are uncorrelated, but if you could knew what all the other relevant variables in the universe were, you could also control for them.  Either way, at that point you could plot the relationship specified in the model by stipulating the mean for all other variables (this would just affect the height of the line).  The other possibility would be to use 'least squares means', better called 'predicted marginal means' to assess a theorized population relationship when all other variables are balanced.  It may help to read How to visualize a fitted multiple regression model? and/or What are LS means useful for?
