they make an income profit or loss on share transactions which is tax accordingly as income. No different to a retail shop buying goods then selling them.
As for claiming GST on share brokerage is not only dependant on the transaction being carried out in a GST registered entity but alos if the share transactions are conducted as a trader or investor.
A trader can claim 75% of the GST on brokerage, an investor can not and the GST is instead added to the cost base or deducted from the sale proceeds.