(1) What are the criteria by which we could determine whether some
field is philosophy rather than mere wisdom?
Since philosophy literally means "love of wisdom", the trivial answer to your question is: there are no such criteria because philosophy is the search for wisdom.
I think what you really mean here is more along the lines: How would someone differentiate philosophy from literature, poetry and from mysticism and religious based wisdom?
In this case there is a clear demarcation line: Religious and poetic dispensations of wisdom are always based on a set of starting assumptions (God, revelation, the nature of beauty and truth, ancestral traditions, etc...) which are taken for granted. Their truth is taken as established fact, and no attempt is made to justify or prove them. "Stealing is bad, because God, the river and trees, the ancestors said so,..."
Philosophy at least tries to justify and argue for its set of starting assumptions, before moving on to giving the details and implications of these assumptions. Take for example Kantian ethics: Kant puts considerable effort into justifying his categorical imperative, and arrives at conclusions along the lines of"Based on logic I described, if everybody were allowed to steal freely, society would collapse, and therefore stealing is bad." In this sense, philosophy is more of a method, a way of thinking, than a topic.
To recap: Philosophy tries to have the least amount of assumptions possible, and tries to argue for the validity of those assumptions using logic and reason, before moving on to dispensing wisdom based on those assumptions. Poetry, Literature, Religion and other such forms of Wisdom don't. Similarly to science what makes a field philosophical or not is based on the method used.
(2) Does it make sense to treat philosophy as merely Western?
Based on my answer to (1): No. Some Chinese and Indian schools of thought, have made legitimate attempts at arguing for and justifying their positions, and went beyond just add hoc dispensing of wisdom and maxims. They have used legitimate philosophical methods, and as such count as philosophy.
Buddhism is a prime example: Although considered a religion, the Buddha arrives at the 4 noble truths strictly through observation and logic. That particular aspect of Buddhism is as philosophical as it gets (even if other aspects of it aren't).