Hi. I've played FullMoon and Gibus quite a bit, the latter of whom I have yet to drop a set to. So I think I can say I have knowledge on how things go in this MU.
Robin's Jab is not the end-all be-all that some would say, but it still is very useful, because it forces Greninja to come in with just one attack: a max spaced Fair. Near anything else loses to the Jab.
Nair? Jab.
Bair? Jab.
A Shadow Sneak off a Jab? Jab 2 it.
Because of that, Greninja's approach options can be a bit constrained. If the Greninja tries to to space the Fair by using a full hop fastfall, Robin has acknowledged it and has his shield up and ready, if not already jumped up and Levin Fair'd you for it. So be very mindful on how you approach Robin.
Good thing for Greninja, then, that Fire Jab is absolutely useless on him. He can SSHC straight out of it. If Robin wants to get meaningful damage off Jab conversions he will have to use his rapid Jab Wind variant and use up a bit of Elwind recovery. "Can't Greninja SSHC out of that one for free though?", I hear you ask. Well, in previous patches, yes he could. Now, however, the Jab 2 knockback buff made it so that Greninja pops too high for an immediate SS counter; if Greninja SS's now, he'll end up too high, and fall right back into the rapid Jab to take more damage. Your best bet is to DI out and away as other characters would.
Robin has a prolonged advantage state, but his means of getting there have commitment to them. While your moves will lose to Jab, your shield doesn't. Jab 1's got a heavy 25+ endlag commitment if he doesn't segue into Jab 2, and that's plenty of time for Greninja to rush up, shield Jab 1, and poke with Dtilt (which Usmash combos into unless Robin's DI game is super on point!) or Jab and get things started. Neutral just becomes a series of 50/50 guessing games on what the Robin and Greninja decide to do. Stay grounded, use your shield well, and pick your battles.
Robin's main spacing tool is definitely Levin Fair, and I don't think Greninja has anything that outranges it. Always respect this tool, from SH, from the ground, from landing, everywhere. 13% fresh and combos into itself, the damage will start to rack up fast if you get complacent. And that's especially troubling with Robin's grab kill confirm. Greninja falls so fast that the % ranges for Checkmate span quite a ways; I've gotten it to work at 90, and work just as well at 120. Arcthunder will guarantee a grab for this, which Greninja can duck under (don't crawl though!), but it doesn't help when Greninja's trying to land, or approach, or come from the ledge, which a well placed Arcfire will force you to act and get off that ledge. If Greninja can keep himself safe from 125 and up, the grab confirm no longer works, but at that point, many other things in Robin's kit will seal it up. Problem there is that the moves are all gonna require a read. Bthrow is an acceptable stock cap at 160 with no rage, Fire Jab would be an incredible stock cap if SSHC didn't nullify it (so don't be afraid of this move close up at high percents!). All of the Levin hits will easily kill at high percents, but with no setups into them, it's gonna be all on the Greninja to mess up with his high mobility to get a hit to connect.
And really, that mobility is what keeps Greninja threatening in this matchup. Robin doesn't really care for Shuriken potshot damage- anything to get Levin Fair roided up and ready to kill fresh at 95 at high rage off a bad misspace is fine by me (Robin can short hop over a high-max charge Shuriken and smack Greninja with Levin Fair in the endlag at anywhere less than max range, btw), but Greninja's pure speed keeps the pressure up while also keeping himself safe from Robin's kit. Just dancing around at mid range with a steady shield propping can make anything Robin does feel like a marital engagement. And again, a max space Fair will outrange Robin on the ground, so Greninja does get rewarded for using his speed and getting his shots aligned correctly. I find Greninja tomahawking to be especially problematic: if he comes in with a move, I'll likely win, but if he just drops down right in front of Jab and blocks, then this is gonna hurt, especially if Greninja successfully grabs from it and unleashes KO grab combos of his own.
Robin has the defense necessary to make Greninja struggle to find himself in, and the kill power needed to make the number of successful blocks and turnarounds needed to KO minimal, but Greninja's speed and mixups can make his own conversions hurt just as hard, provided his shield game is potent and he respects Levin Fair. This is a matchup where Greninja needs to find ways to wallbreak, not play footsies in.
50:50