Yes, the buyer can bid on both, you end the auctions immediately, and he is deemed to be the winner at his opening bid.
That is , if your opening is $10 and he bids $100, he wins it at $10.
Better would be to make a private Listing (not auction) for him.
Put the pictures and descriptions into the Sell Your Item form --just cut and paste.
Set the price as agreed with the customer.
Adjust the shipping cost.
Make the title his eBay ID.
Make it Immediate Payment Required. (One good reason for a Fixed Price listing).
Tell him the Listing is up , the title, and that it is only good for X *days.
He purchases, he pays, you ship.
Since this takes less time than your two auctions (or does it?) you might take a chance and leave those up for other bids. If he wins you cancel the bids and end the auctions which go to your Unsold List for 60 days. But I wouldn't take the chance.
If he doesn't pay within the time you gave him, Block him for a timewaster and relist your auctions.
EBay is not really an auction site anymore.
Over 80% of transacitons are Fixed Price and even on Auctions most have a Buy It Now option.
This is a reaction to mobile shoppers and to impulse buying.
Think about it.
*The cheapest Fixed Price listings are 30 days. But give him much less. Two days tops.
