Aspect of the monkey will take that lion animal companion beyond what normal lions can do; that's what it's for. Still, I've commented before about the odd combinations of climb being based on Str and the highest Str animal companions. If lions going up ladders bothers you, wait until you see an elephant doing the same.
The Lion in question is not your typical Lion anymore.
A wild Lion is not expected to make such things possible... no. In a fury, a wild Lion may follow you up a ladder, or out of desperation/starvation figure out how to climb a ladder... but you your average beasts are not intellectually capable of immediately doing these things.
The Hunter's companion is different. It is designed to be a tactical companion. Giving a Lion the climbing aspect of a monkey is an advantage wild Lions do not have...
The Hunter is smart enough to strategically gift his Lion with the know-how of monkey climbing, and I would have to infer that the Lion still knows how to hunt on its own and incorporate the new abilities into its tactics.
The Lion immediately recognizing that a ladder is capable of being climbed? Absolutely yes, especially if the Hunter can also see the ladder.
A vertical cliff face? Sure, they get the bonus from the Monkey Aspect, but the checks have to be made, regardless...
Mufasa didn't make his Climb check...
You can always have something with Grab & Swallow Whole just eat the stupid cat in question with an attack of opportunity.
They can do it if they can make the check.
Alternatively, most animals can't climb at all:
You need both hands free to climb
But the skill rules are written in a humanoid-centric manner, so I wouldn't read too much into that clause. (For that matter, it's a very stupid clause, as plenty of folks climb one-handed.)