I hate to be the bearer of bad news, but I can tell you with absolute certainty that this is *not* a WWII P coat. In fact (and I'm sorry to tell you this), it's not even close to the WWII period.
From this one picture, it appears to me an 80's-era piece, though I can't be 100% sure without seeing several of the coat's details.
If the gift-giver paid a premium price for an item that was represented as WWII vintage, they should contact the seller and get a refund; the seller was either preying on the ignorance of the buyer, or ignorant themselves (as is often the case). Late-model coats like yours can be had very cheaply on eBay; usually under $50 for an item in excellent condition.
That said, if the buyer didn't pay through the nose for it, and you don't have your heart set on a true WWII item, keep it! Although pre-war and wartime coats are warmer, better constructed, and made with superior materials (heavier, 100% wool, as opposed to the lighter, 80/20 wool/synthetic blend from which yours is made), you still have a very good garment. It will wear harder and longer than your average designer-label "peacoat," and unlike many couture items, the cut of your coat is absolutely timeless; 50 years from now, you can wear it and be perfectly fashionable.
I can probably date your coat down to the year, but you'd need to provide me with some more pictures. Let me know if you're interested.
PS: Removing the patch will be very simple; tailor, or a seam ripper + patience will have the same effect. Either way, I'd say remove it. Possible disrespect-to-military-personnel aside, feel the patch not only detracts from the timelessness of the coat, but also takes ones attention away from the coat as a whole.