Why are shares exempt from GST? Why is the financial trading sector exempted from consumption taxes?
For example, shares seem to be exempt from GST in both Australia and Canada. What is the intention behind this? Is it common for jurisdictions to have a VAT or sales tax that also covers or excludes financial instruments and supplies?
Semantics aside, is there something that would make it infeasible for some kind of sales tax to be levied at an identical rate on every transaction in the economy?
Shares, and many other financial instruments as well, are financial assets  assets, through which people save not consumption of goods and services. Consequently, neither GST (goods and services tax) or value added tax (which is in economics a type of consumption tax) can apply to stocks or many other financial instruments as they do not represent consumption of either goods nor services. Hence prima facie there is no reason why consumption taxes should apply to things that are neither consumption or goods or services in the first place. That is like asking why houses are exempt from labor taxes... well its because they are capital not labor.
For example, if you pay someone to make trades for you (and hence someone to provide you financial service), you will generally be required to pay VAT or GST on that (unless particular country decided to exempt it for whatever political or legal reason - which is beyond the professional competency of an economist to comment on).

Response to edit:
It is feasible to tax financial assets but this would be considered wealth taxation (completely different type of taxation than consumption taxes). Taxing wealth is feasible in a sense government can pass legislation to actually do it, but from economic perspective it is often considered undesirable for its negative impact on the economy.
The literature shows that optimal top marginal wealth taxes, even in case of Rawlsian social utility function (the most redistributive social function there is), are somewhere in range 0-10% (see Kocherlakota 2005; Fama 2019; Zucman and Saez 2019 or this article by Summers) or discussion in the famous Mirrlees Review: Dimensions of Tax Design). Also, note the consensus seems to still be closer to the zero result and only very few scholars would go as high as 10%.
The above being said governments typically do not conduct science based policies. Often politicians will ignore even recommendation of their own analysts when it comes to the design of tax policy. Consequently, it is beyond scope of economic profession to tell you why politicians do not base all their laws on best published science/state of the art models or why they do tax one thing but not another (for example, in the Netherlands where I live for long time food for hamsters and bunnies was taxed at different rate even though they are perfect substitutes - which is completely absurd from any economic perspective, yet that is what the government legislated).