One of the two sets of shares will be sold, obviously - either the one you had before, or the one you get from exercising your call. And yes, this can have impact on your taxes, potentially triggering a capital gains tax.
However, most brokers allow you to specify which 'bundle' of shares you'd like to have sold - typically for one or two days afterwards (but it must happen before settlement). So simply log on to your account after the trading and exercising happened, find the respective transactions in the log, and edit the affected share block to your liking.
If your broker doesn't have this option on the website, call them (but again, before settlement!), and ask them to adjust it - and maybe change brokers.