What is the purpose or message behind the dog figurine that Bond receives?

Well I would choose not to speculate on the Bond-M relation and rather point out something from the movie.

Cut to the scene where Moneypenny hands Bond the object in question:

Moneypenny: Maybe that was her way of telling you to take a desk job.

James Bond: Just the opposite!

So, IMO, that was M's way of providing an emotional nudge to Bond, to carry on, from the far and beyond.

The dog represents Bond and the hate/love relationship between him and M.

M feels responsible for turning Bond into the cold, detached (ugly) agent that he's become, and that, combined with the fact that he's so good at it, is why she refuses to let him go. Hence, why the dog is so ugly and why she insists on keeping it.

Bond, on the other hand, would rather be somewhere else, but, at the same time he feels that it's his responsibility to protect M (which is the reason why he returned from being "dead"). This is why he hates the dog, and why he'd rather her threw it out. This is also the likely reason why the figurine is a dog and not something else. Just like a dog, Bond will stay by M's side until the end.

The dog surviving the explosion makes this even more obvious. It, just like Bond (and his relationship with M), seems to survive pretty much anything.

Leaving the dog for Bond is M's way of not only getting the last word in their never-ending quarrel, but also to remind him that, while he might be an ugly dog, he's UK's ugly dog (the fact that the dog is an English Bulldog draped in a UK flag indicates that it's not just about M, but the entire country), and they still need him despite that she's now gone.

Bond isn't fond of the "British Bulldog" figurine, but clearly M is. He makes disdainful remarks when he sees that, of all things, the ugly figurine he dislikes so much survived the explosion.

When he receives it at the end of the movie, it's because M left it to him. She knew how much he disliked it, so what could be a better present to remind him of her!

"After World War I patriotic bulldogs draped with Union Jacks were introduced into the Royal Doulton collection, because the bulldog was a symbol for the dogged determination of British people. The statues were reintroduced during World War II." It simply urge Bond to keep doing what he does with his dogged determination & for the people of the British Empire. Also since he dislikes it most & it survived the explosion at MI6, tells him even to outlive M, to keep on going.

I simply thought of it as being a collegial joke by M, because Bond wondered a bit sarcasticly why solely that dog had survived the bombing. Just as in: "Oh, you don't like it? Have fun with it then!"

Bond doesn't like the dog, we know that from the scene in the film. M leaves Bond the dog because she knows he doesn't like it. Bond then says something along the lines of, it means the opposite of get a desk job. My thought on this if he got a desk job he'd have to look at the dog all day!

"In the opening sequence, when M is writing Bond’s obituary, the bulldog is facing outward as if M is watching England. Throughout the rest of the movie, the bulldog is facing M, as if England is watching M. When she wills the bulldog to Bond, she’s giving him responsibility for England." Source

Yes, the bulldog could be a reference to the world war days and the determination of the Brits, especially with the mention of Churchill's tunnels; however, the overall underline reason for the dog is because M and Bond are both old dogs in a new world. Bond doesn't like the dog because it reminds him of this. When Bond first returns after his death, M and him have the dialog, "You know the rules of the game; we've both been playing long enough." To which bond replies, "So this is it; we're both played out." Bond and M are cut from the same cloth, kindred spirits, a throw back to an earlier time.

This is echoed when Eve gives Bond a close shave and says, "Old dog, new tricks." This is also a reference to the reboot of the Bond series: an old character being presented in a new way. Which is further explored in the introduction of a new pimple faced Q. Exploding pens, "We don't really go in for that anymore." "A brave new world," Bond replies. And then when Bond goes old school with his gadget car, he return briefly to the old Bond ways (but still done in an updated way).

It represented England in the beginning of Skyfall when M was writing Bond's obituary. The Bulldog was facing outwards as if M was looking out for and over England. For the rest of the movie, the dog was facing inwards as if England was watching her. Then at the end when Eve gave the dog to him, it was M telling Bond that now she wants him to look out for and over England...

There is a common cultural crossing between the bulldog and Winston Churchill (so much so the Churchill insurance company play on it in adverts).

In 1940 early in the Second World War, Winston Churchill had had a long service, but had been exiled due to his views, including his "lone voice" on the dangers of Nazi Germany and the coming War. However once he returned to become Prime Minister in 1940,it's widely felt that his singular effort was a deciding factor in the winning of the war (and why as a commoner he received a State funeral).

The parallels are there. Bond is in exile after he "dies", he has lost faith in what he does, the politicians are trying to get rid of the human element (the double o section), but his lone wolf approach is what defeats Silva when the might and tech of MI6 fails.

The message to Bond is clear, although you may question what you are doing, and others feel you are an anachronism, your country needs you now and you are the lone protector. As Bond says when Moneypenny suggests it means take a desk job

Quite the opposite.