What was Bruce Wayne's obsession with unusual looking statues?

These are not just statues, they're suits of armor and battle-garb from various cultures. Many of them are European plate armor (basically all of the full-metal ones). The fourth image's is Japanese. I'm not sure where the armor in the fifth image is from, exactly, but I suspect it's one kind of African or Native American garb (if someone knows what it is specifically, please do comment). The twin suits in the last image are again European.

They're a reflection of him being a warrior, one who has traveled the world learning many styles and philosophies of fighting. He his own suit of armor which follows the trend of the ones on the pedestals- they are both protective and intimidating.

They may have some personal significance to Batman, but the main reason for it, both in and out of universe, is to set up a distance between Batman and anyone that sees Bruce. As in any Batman medium, Bruce is a playboy. So filthy rich it's a joke. Extravagant. Expected to be odd and eccentric, as rich people are known to be. It's explicit in how Vale and Knox talk about Bruce. It's how people talk about Clooney's, Bale's, even West's Bruce. West had abnormal tastes. Clooney had the garage. Bale burned down his house at a party. Followed by their very carefree use of money. Clooney's donation to the Circus, Bale's purchase of a restaurant to get a seat, and Keaton's generosity in opening cases of wines and giving Knox a grant...

It's all part of the act, the facade of Bruce Wayne.

From the 5th draft script essentially what was in the final:

                             KNOX
                            (ironic)
               And here we are in the arsenal.

    On the wall hang EXOTIC WEAPONS.  Every arcane implement
    of death the human mind has ever devised.  Knox lets out
    a low whistle.

                            KNOX
              Look at this stuff.  Who is this
              guy?

    Bruce ENTERS and stands just behind them.  Vicki sees
    him, but having seen him earlier, regards him casually,
    like another partygoer.  Bruce listens with interest.

                            VICKI
              Strange.  He gives to humanitarian
              causes... And collects all this...

                            KNOX
              Probably does it to get chicks.
              They like him for his big charity
              balls.

                            VICKI
                     (mischievous)
              I think it's his enormous...
              bankroll they go for.

                            KNOX
              Hey, the more they've got, the
              less they're worth.

                            VICKI
              This guy must be the most worthless
              man in America.

Later on:

                            BRUCE
              Oh yes... thanks.  Oh, Alfred,
              they need mare wine in the front
              and someone named Mrs. Daly wanted
              a copy of the menu.  Oh, and,
              Alfred.  Give Mr. Knox a grant.

    He winks at Knox. And goes the OTHER way out of the
    roam. Alfred fluidly sweeps up Bruce's glass and follows
    him briskly OUT OF THE ROOM.  Knox is stung and Vicki
    transfixed.

                            KNOX
              Nice talkin' to ya, Bruce.  Now
              are the rich odd? Yes they are.
                     (jealous)
              Helllo?  Vicki?

                            VICKI
                     (waking up)
              Sorry, I was -- He's comp-lic-ated.

                            KNOX
              I said odd.

                            VICKI
              Mm.

                            KNOX
              Ah -- Well you're not the only fan.
              This guy loves himself.  There're
              mirrors in every roam.

    And indeed, the two of them are standing before an
    enormous WALL MIRROR, eight feet wide, running from
    floor to ceiling.

                            KNOX
              Maybe it should be Bruce VAIN.

Batman is pretending and faking as needed to keep them thinking of him as he wants them to think.

Batman is also prone to collecting. He keeps trophies of everything he does. His cave is full of weapons and devices meant to kill him. The Penny, the Dinosaur, etc. All of his different suits, and the suits of his robins, in a case, on display. You could say it's an orphan or neglected single child's obsession with filling a hole in their heart with stuff...

I agree with @HarrisWeinstein that they are suits of armor, and the batsuit is Bruce Wayne's "suit of armor". But they also look intimidating: some have deadly spikes sticking out, one has terrifying protruding eyes, another has "wings" and a death grimace. The batsuit is designed to intimidate batman's opponents; that's why Bruce Wayne chose bats in the first place, instead of just having body armor and a mask. He wants his opponents to be scared of him.

I have always seen this as foreshadowing of Batman as an armoured but somewhat fantastical figure and they could be seen as Bruce Wayne's inspiration for the bat suit.

It is worth noting that while they are clearly based on real wold armour they all have a bit of a fantasy spin. For example looking at the smaller images from left to right the first is a bit like European late medieval armour but a bit too bulky and the spikes are pure fantasy and go completely against the concept of how this type of armour works. The second looks vaguely like a WW1 gas mask but the helmet is obviously modern (looks a bit like a British Mk6 or para helmet with extra bits attached). The third is vaguely Japanese and I don't have the knowledge to say that it is definitely wrong but I have never seen anything remotely like it.

As far as I can tell the final one is pure fantasy.