The Knowledge passive works very similarly to Catchup.
On a technical level, it just adds 0.05 to 0.15 to the same Exp multiplier.
It doesn't multiply with Catchup, and it also doesn't apply very often, just as Catchup— only applying on battle-related Exp on your main team.

Many people have thusly found the Passive to be completely useless, and while I do agree to a certain extent, the passive is certainly difficult to use:

• Using Knowledge instead of a battle-oriented passive will certainly grant lower Streaks, and since better Streaks grant better Exp, Knowledge can definitely hurt your Exp efficiency in some cases.
• The stat range of the passive is large → a low quality passive won't be worth the effort.
• Many people streak for flex, not for Exp.
• Building a functioning team around having effectively 1 less passive can be difficult.
• Knowledge doesn't multiply with Catchup → to use it well, you'd need to give it to a pet that doesn't receive Catchup.

I don't believe most players give it it's due consideration. It can actually be worth the effort if you're optimising your team for Exp-effectiveness, as most players are.

• If you have a team of similarly leveled pets, then using Knowledge on your highest leveled pet basically multiplies the leveling speed of your whole team, just by using the passive once. This is because of the way that Catchup trickles down to your other pets & always keeps them roughly at the same level.
• A 1.15x leveling speed for your entire team is honestly huge, lol.
• Some teams don't rely a lot on each passive. The Sword passive on Double Gem AOE for example is unimportant and you can easily run Knowledge permanently there without issues.
• And then there's the option of swapping to a Knowledge weapon on Streak Milestones.. you basically have no losses in doing this, since the risk of losing your streak in that single battle is miniscule. So why doesn't everybody do it? Just because it's pretty annoying to have to switch weapons. But otherwise? It's super-duper effective and safe.

The Desmos link I've added to NeonUtil Global above gives you your Battle exp. Feel free to do the calculations yourself, whether the passive is worth your time! I hope I'm spreading my enthusiasm for math.

I won't be talking about Knowledge in Weapons & Passives, I feel like these paragraphs here are sufficient.