It seems wrong to admit that a Death Eater posing as an auror was probably the second most capable Defense Against the Dark Arts teacher that Harry ever had, but it's still true.
While, technically, fans spend far more time with "imposter" Moody in the story (when Barty Crouch Jr. was disguised as the auror) it has to be assumed that this was a very close representation of who the real Moody was or, Dumbledore at least, would have noticed the differences. But, even including Barty Crouch Jr.'s version of Alastor Moody, there were quite a few things that never quite made sense about the character overall.
Getting Kidnapped
The fact that Moody was able to be overcome, kidnapped, and imprisoned for nearly a year doesn't make sense with the reputation the famed auror was supposed to have.
He was supposed to be, perhaps, the most famous auror ever. He was in charge of the department and mentor to many of the up and coming dark wizard catchers. Yet, he was able to be overcome at his own home by one man alone, Barty Crouch Jr. Doesn't really hold up when you think about it.
Seeing Through Invisibility Cloaks
Readers of the Harry Potter books know that Moody is able to see through invisibility cloaks with his magical eye. While not enough is known about Moody's magic eye to really judge all of its abilities, the fact that it is able to see through Harry's invisibility cloak, in particular, is strange.
As fans eventually find out, Harry's cloak is not a traditional invisibility cloak, but one of the Deathly Hallows, and especially magical. That Moody was able to see through this invisibility cloak is strange.
Not Trusting Snape
Moody was never a particularly trusting soul, especially where Death Eaters were concerned. The fact that he put so many Death Eaters away only added to his animosity.
However, with his magical eye, his trust of Dumbledore, and his uncanny ability to sniff out the truth and wrongdoers, the fact that he couldn't dig up the truth about Snape is weird. If anyone could have known enough to trust Snape's motives it should have been Moody.
Tonks Was His Protege
Of course Tonks worked under Moody in the Auror office. He was her boss and undoubtedly taught her many useful skills on the job. The idea that Moody took special interest in Tonks and taught her especially it harder to believe.
Tonks was a good auror, but she wasn't Moody. She was young, clumsy, and had a sense of humor that Moody didn't always appreciate. It seems more a matter of convenience that she should be his protege, rather than a meaningful character choice.
Order Leader in Dumbledore's Absence
Dumbledore always had a lot on his plate, including running a school. When he was otherwise occupied Mad-Eye was in charge of the Order of the Phoenix (just see him leading the Advanced Guard to get Harry away from the Dursleys) After his kidnapping and imprisonment, as well as his prickly personality, it seems like an odd choice to leave him in charge of everyone fighting the good fight in the coming Second Wizarding War.
Perhaps there was no one else, but it still doesn't quite make sense.
No One Knew Him
Again we come back to the fact that Moody was kidnapped and imprisoned for over a year and literally no one noticed. Sure, his auror buddies weren't working with him at Hogwarts, but they didn't check-in? They didn't visit or write? What about his protege Tonks?
The fact that no one missed him or knew him well enough to notice that something was wrong when he showed up at Hogwarts is a hard pill to swallow. Everyone seemed to have a history with him once he reappeared in Order of the Phoenix.
Harry Didn't Have Hard Feelings
Of course the "real" Moody didn't offer Harry up on a silver platter to the Dark Lord, but his doppelganger certainly did. This doppelganger was enough like the real Moody that no one noticed the deception. The fact that Harry was able to separate his experiences and feelings (despite the PTSD suffered after the Voldemort incident) is odd.
Moody wasn't a kind and fuzzy enough person to really encourage Harry to build the relationship he was able to in the end.
Unforgivable Curses
It was Barty Crouch Jr., not Moody, who showed fourth-year students the unforgivable curses in class. Yet, Dumbledore didn't think this was strange which means it's also something the real Moody would have done.
This is a strange detail for a man who couldn't even forgive or trust rehabilitated Death Eaters. Constant vigilance, sure, but displaying the evilest spells in the world to students seems a bit out of character.
Body Never Found
Moody was murdered by Voldemort himself when spiriting Harry away from Little Whinging before the boy's seventeenth birthday. The famed auror's body was never found.
Was he disintegrated? What about "accio"? Odds are he plummeted to his death over a muggle neighborhood, which certainly would have caused a very public stir. It's hard to believe any of the Order looked very hard if they weren't able to find him.
Let People Call Him Mad-Eye
Sure, it's an intimidating name that likely thrust fear into the hearts of the dark wizards that Moody hunted, but it's weird that his friends and colleagues would also call him "Mad-Eye" rather than Moody, or by his first name, Alastor. It's weird to call attention to one of his war injuries, even if it is particularly magical.
Moody was no complainer, but it still doesn't quite make sense that he would let the nickname stand.