There's a general consensus about Hemingway having been something of a fantasist — a master storyteller who frequently blurred the lines between fact and fiction in the name of great narrative or just plain old ego. Literary scholar Edward Mendelson puts it well in this review of a volume of Hemingway's letters:

Hemingway’s letters are copious with gossip and boasting that he often enlivens by invention. He assures a friend at one point, “Quite a lot of the above paragraph is true.” Except in a few authentic-sounding outbursts against his parents, almost everything he says about his emotional life seems false.

So I'd take that into account when reading that execution anecdote, or his claim to have killed hundreds of Germans. There has been a lot of debate about Hemingway's wartime activities. There's an excellent essay (Correspondent or warrior? Hemingway's murky World War II "combat" experience) by journalism scholar William E. Cote, which reviews the scholarship on Hemingway and his actions in France in 1944-45 and attempts to sift through Hemingway's own conflicting accounts of his combat experience.

Cote's conclusions are, broadly, that most of Hemingway's combat anecdotes are unverifiable and most likely untrue (or at least heavily exaggerated.) He includes a particularly telling quote from Time-Life correspondent Bill Walton, who spent time with Hemingway in France:

He was a storyteller. He didn't know when the truth and fiction stopped. And every now and then, just when we were together, he'd tell me a story, and then I'd look sort of strangely, and he'd say, "You don't believe me, do you?" [Laughter] And I'd say, "No, I don't." He'd say, "You're absolutely right." [Laughter]

In answer to your broader question: after D-Day and a sea change in Allied public relations policy, war correspondents on the Allied side had a lot more autonomy/freedom of movement, and some, like Hemingway, took on a strange — and often completely inappropriate — role. The mid-20th century golden age of war correspondence (the Spanish Civil War and the Second World War through to Vietnam) and the experience of those journalists is often characterised by macho adventurism.

For a good read on journalism in war, check out Phillip Knightley's The First Casualty (you can read it online at that link.) It does a pretty good job of dissecting war correspondents' complex roles and competing motivations. Here's Knightley on the correspondents, including Hemingway, who accompanied the Allied advance through France and into Germany:

Some correspondents began to carry guns, and the American and British armies seemed to be preceded at times by jeep loads of journalists, who frequently found themselves taking towns and accepting surrenders. Ernest Hemingway, whose style of war reporting was to take part in the action as if he were an infantry officer and then to write about what had happened to him, reported that he had managed to gather a small combat outfit of his own, mostly French irregulars, and get into Paris almost a day before the official liberation.

(It's also worth noting the turn of phrase "Hemingway [...] reported that he had managed..." because that story about him being the first into Paris — according to biographer Jeffrey Meyers and others — isn't actually true either.)

That quote also doesn't tell the whole story, which is that these kinds of paramilitary actions were completely unsanctioned by the military command and created serious tensions between the generals and the press, because journalists were regarded as noncombatants; their taking up arms was potentially a violation of the Geneva Convention, and at the very least a violation of US Army regulations for war correspondents.

Hemingway himself was investigated by the US Army's Inspector General for alleged paramilitary actions. Per Cote: "James Meredith has concluded that Hemingway lied to the Inspector General in denying he had armed himself, commanded fighters, and covered up his war-correspondent's insignia." But how far did pretensions of command translate into actual military action? We don't know — but it certainly wasn't dramatic a campaign as he claimed.

The overall appearance of Hemingway's wartime 'combat' experience is somewhat similar to his abortive espionage career in the early years of the war — when he attempted to develop relationships with at least four US intelligence agencies and the Soviet NKVD — personal ambition mixed with delusion, and heavily embellished in the later telling.

Hemingway scholar here (MA). Ernest was prone to flights of fancy with respect to his latter-day military service. There is no question he served bravely in Italy with the ARC Ambulance Corps. Neither is there any question of his willingness to endure danger for the purpose of being in (and consequently getting) the story. His involvement in the Spanish Civil War was as a correspondent for NANA (the North American Newspaper Alliance). He reported on WWII for Collier's Magazine. He also 'patrolled' Cuban waters for German subs in his fishing boat the Pilar.

Everything Hemingway wrote was in some measure autobiographical. He & his friends were clearly identifiable in his fiction, though being a writer he made them more loathsome, more beautiful, more brave, or more foolish than their real-life inspirations may have been. These embellishments are the stuff of authorial license, and often came mixed with a healthy dose of passive-aggression. If he was willing to run his friends, lovers, and mentors down in print (e.g., Fitzgerald, Stein, Anderson) for faults their generosity to him apparently could not obscure it stands to reason he would find equal ease in gilding his own lily, so to speak.

Hemingway knew he was a star. The stories had to keep coming else he wouldn't be a star anymore. When he finally successfully shot himself in Ketchum, ID after repeated suicide attempts many biographers attributed his act to a combination of repeated TBIs (the history of which is fascinating in and of itself) and depression arising from fears about his lack of successes in print (he had a long drought before 'The Old Man And The Sea' won him the Nobel Prize for Literature) and the sort of normal health problems attendant on aging men (e.g., impotence, painful skin condition). He had ceased to see himself as a star and felt like a fraud. In some sense, his expectations of public expectations of him drove him forward into glory (sometimes completely imagined) until he realized he wasn't going to be able to do it convincingly anymore. In my opinion and in that of most Hemingway scholars, stories of EMH killing Germans are a component of that embellishment.

Note: Sorry about all the parentheses. Hemingway scholars write entire books about single sentences of this post. Please excuse me for discarding propriety in an effort to pack this post with information & context.