Why wouldn't they be able to? That's what the skill does.
"No, as true norsemen, they must kill and eat every animal they see".
Seriously though: you can have a pet. But just using handle animal alone isn't very...good, honestly. There are plenty of restrictions.
Unlike animal companions, which have a special ability affecting handle animal checks, 'pets' that you just buy or catch will require move actions in order to command. Also note: they do not scale at all. So a good pet now is near useless later.
Also, there are other restrictions, which could be reasonably used by a GM to settle the issue. The first and most important one: training an animal to do a 'trick' (ie: following a single kind of command) requires 1 week of training. If your campaign doesn't have time for you to sit around for 6 weeks... well, too bad. Note: Proper animal companions don't have their problem, since their bonus tricks come pretrained when you get it.
Other considerations: while a tiger might seem cool... I can understand your GM denying it. It is a funny thing about animal prices: it was more flavor, so technically a level 2 character could buy a 4 HD embodiment of pouncing death. So... yeah, taht is a balance issue. Getting a watch dog is one thing, but getting an entire boss fight as a pet can be troublesome. So I can see the GM reasonably restricting you to rather... watered down selection of pets. IE: you can't get a tiger until after everything else is powerful enough that the tiger isn't actually good anymore.
If you want an Animal Companion and not just a pet as described above, you could invest in the Nature Soul --> Animal Ally --> Boon Companion feat chain. Ways to get an Animal Companion using VMC exist, but they tend to be rather bad, eating 5 feats instead of just 3, and coming online later.
I haven't yet heard of a Skald archetype that gives you an Animal Companion (that is, unfortunately, no Skald equivalent of the Mad Dog Barbarian is out yet).
Yeah, animal ally is probably better in the long run. The companion is usually scaled so that it is somewhat appropriate. Neither being several hit dice above you, nor being completely obsolete in a couple of levels... once it comes on with boon companion at level 7.
Also, it involves the expense of some actual resources. Not just a mere 300 gp, but 3 darn feats.
The only real problem I can see is that it is the ranger list. So not tigers or bears... but you can get a wolf as big as a horse at level 7. It can deal pretty decent damage too once it gets the extra attack from multiattack, along with buffs from your skald.
As lemeres mentioned, buying more powerful animals out of the book might be something some DMs object to for game balance reasons.
Getting an animal companion via feats could be a good choice. If it is just a pet rather than a mount it might be cheaper to pick up a Mauler familiar. Either way, your raging songs should help increase the creature's combat effectiveness.
Depending on how your DM feels about summoned animals you might be able to get a lot of use out of Handle Animal and possibly even Ride with your Summon Monster spells. This starts out kind of weak, but some of the higher level summons could be great allies or mounts, especially with raging song.
I think it would be cool if high level PCs could grant temporary HP to their mounts, perhaps with something based on your Handle Animal and or Ride skills. That way even somebody who hasn't played a special class or taken 3 feats could still participate in mounted combat at higher levels without having their horse shot out from under them almost immediately. I guess being able to use Mounted Combat to substitute a Ride check for the mount's Reflex save might help too. Of course anybody with that feat likely has a class feature mount too.