I don't know what to make of the first scope shot. Is it aliased? It seems much too periodic for that though. Seems like burst-mode operation under light load conditions, but the ramp is wrong. That would be correct if the load is really nonlinear. Wouldn't make sense that it's starting up just a little as the voltage falls, and then suddenly kicks up to full power and overshoots. Would it?...
Or maybe it's from uncompensated scope probes? Does the square test waveform on the scope look correct, or a bit rounded or peaked? Adjust the trimmer screw on the probe (body or plug) to flatten it out. (Shouldn't be the same time constant though, that's normally in the ~1ms range, not 100s.)
The second one looks like normal loaded ripple. It appears to be mainly V = ESR * I_L. You can tell because it's mostly triangular, as the inductor current is. A purely capacitive load would integrate the triangle wave into a quadratic wave (which the datasheet calls a "sine", but it's subtly different from that).
Did the datasheet/devkit not use an aluminum polymer originally? Electrolytics do come in some nice ratings, but I don't think there are all that many, in that value, to choose from. Similarly with tantalum, for which you made two mistakes I suspect:
1. Tantalums come in a range of ESR. If you picked one at random, chances are its ESR is on the modest side (~1Ω)? If it's rated anything at all; beware AliExpress!
2. Tantalums lie about their voltage ratings (even from major brand names). They
might survive at rated value, but your chances are greatly improved with a 2-3x derating. Yes, a 35 or 50V part would be safer here!
Don't take this as blame -- you probably haven't heard about these things before.

Also consider ceramic capacitors, if the controller is up to it.
Looking up the datasheet...
Judging by the block diagram, LM5164 is a hysteretic controller (note the feedback
comparator, not a linear error amp with compensation), so you can't filter the output too heavily. The C value matters, and should be modest, not huge. And ESR will be a part of that.
Probably if you use ceramic capacitors, a lower value is needed -- to keep Vripple in a reasonable range for the controller, so the capacitor impedance should be kept similar. Point being, ceramics have really low ESR (like 10s mΩ).
In fact, I see the recommended application uses ceramic; and indeed the calculation suggests like 3uF, but they use a 22uF ceramic, and they explain this is because of the quirk of type 2 ceramic capacitors, which is that C varies with V.
And no, you can't really avoid quirks on components, everything has something funny. ESR, ESL, EPC, there's no perfect component, only approximations thereof. What you're really doing, is making sure the approximations match, in the ranges where they are applicable (over a range of frequencies, or voltages, etc.). Which is usually okay, but you can get tripped up from time to time.
So, you should shop for a capacitor that has a similar impedance, at Fsw(nominal) and Vout, to what's calculated by the formula. An electrolytic or tantalum may well be 100uF or more and will be essentially ESR providing that impedance. A ceramic cap can handle the ripple current, with smaller values, while offering a more capacitive impedance.
Shopping for ceramic capacitors is awkward, because no one reports C(V), you always have to dig through datasheets or online databases to find the curves. If they have them at all, which if they don't, is a big no-sale for me.
High frequency (switching) noise depends on ESR and ESL, so a ceramic will tend to let through less than the alternatives; but if switching noise is a problem, a second stage of filtering (on both the input and output) is probably called for, anyway. In that case, just a little inductance (0.1-1uH say) and capacitance (~uF) is needed.
Good luck in school. It sounds like you're way ahead of, easily 90%, maybe even 99% of your future class. You have a lot to learn I'm sure, but you are well prepared for the first year or two if you're taking on subjects like this already.

Tim