I stumbled on the same discrepancy, and was puzzled by a significant difference between the two prices on ETR and FRA. For example, today is Sunday, and google shows the following closing prices for DAI.
FRA:DAI:
Date Open High Low Close Volume
Nov 20, 2015 78.55 79.50 78.50 79.36 8,910
ETR:DAI:
Date Open High Low Close Volume
Nov 20, 2015 79.16 79.60 78.60 79.25 3,076,967
So it looks like there are indeed two different exchanges trading at different prices.
Now, the important value here, is the last column (Volume
). According to Wikipedia, the trading on Frankfort Stock Exchange is done today exclusively via Xetra platform, thus the volume on ETR:DAI is much more important than on FRA:DAI.
Obviously, they Wikipedia is not 100% accurate, i.e. not all trading is done electronically via Xetra. According to their web-page, Frankfort exchange has a Specialist Trading on Frankfurt Floor service which has slightly different trading hours.
I suspect what Google and Yahoo show as Frankfort exchange is this manual trading via a Specialist (opposed to Xetra electronic trading).
To answer your question, the stock you're having is exactly the same, meaning if you bought an ETR:BMW you can still sell it on FRA (by calling a FRA Trading Floor Specialist which will probably cost you a fee).
On the other hand, for the portfolio valuation and performance assessments you should only use ETR:BMW prices, because it is way more liquid, and thus better reflect the current market valuation.