One problem in using exchange rate when comparing GDP per capital between countries is that is fluctuates a lot. A way of avoiding dependence on exchange rate is to use purchasing power.
GDP is a flow!
Lastly, note that GDP is a flow variable, not a stock variable. By a flow variable we mean a variable which is measured in something per unit of time. If you fill a bath tub you may fill it at 40 liters per minute - a flow - whereas the tub itself may comprise 200 liters - a stock. Similarly, income is flow (you may make 9 euro per hour) whereas the amount of money you have in your bank account is a stock (you would never claim that you have 2400 euro 'per month' in your account - you have 2400 euro period).
GDP, being a flow, isn't a measure of the total wealth of a country however a measure of the 'income' of the country during a particular period of time. Sure if GDP is high, it is very likely that total wealth of the country is increasing over time (some wealth is lost to depreciation). Consequently there is often a connection between what we perceive as a 'rich' country and a high GDP per capita.