I don't know much about Morty but apprentice-era Mort was (inadvertently) invulnerable while he was still only an assistant to the Duty and when he was the 'supply Death'/Death's stand-in he did at least as much as Dᴇᴀᴛʜ occasionally does himself to deny Fate, purely by accident. I mean, the latter has connections (all the way up to Azrael, as we see in Reaper Man) and influence and probably
can do much more (practically anything other than 'process' Rincewind), but the devotion to the Duty keeps Him mostly on the straight and narrow (happily letting others do "what is necessary and right, but not permissable"), give or take the occasional swarm of flies being subject to his (early-installment weirdness) annoyance about being foiled by a mere Wizzard.
I don't know what Morty has going for him, but I'd be more definite about there being awkward questions about who'd 'win' between on-Duty version of Mort and, say, Bill & Ted. Or
Gary Gygax. Though I suspect Mort would have an edge that neither Deaths involved in those did, much as he acts against his own master's nature in causing/handling the problems he is initially responsible for.
In-universe, it seems that we never saw Cohen The Barbarian have to deal with more than the maidens that would convey him to his final 'reward', and he bested
them, but not sure about the scythed-personification. I do think the only thing that would have stopped Cohen from similarly getting Binky from Mort would have been Binky, though!
Pre-apprenticeship, Mort can't even scare crows, of course. And post-apprenticeship he seems to do well as duke (he at the very least helps arrange[1] for the peace and prosperity only previously guaranteed under the ægis of the ambitious and ruthless high-flying strong ruler that was previously Fated to take power over the whole expanse of lands and cjty-states), but he is proven to be mortal[2], however competent he might have been.
Briefly testing the waters about what (Rick and) Morty's dominant qualities might be, it looks like he has a many-world multiplicity of selves to call upon. Which
might work out well against any version of Mort short of the Binky-riding and scythe-wielding instance. Shorn of all plot-armour and hypernormal associations, it also seems like baseline Morty and ignored-by-crows Mort are potentially competitive in the very different "think too much, achieve very little" stakes. But once you start moving into competitive 'full hyper-competent potential' territory, I would tentatively opine than one couldn't even match a mind-control eyepatch against Tʜᴇ Vᴏɪᴄᴇ... Drawing upon my familiarity with Disc matters and my probably far-too-rapid scan of the non-Disc ones, anyway.
[1] Or assist, in his new position as a duly ennobled ducal subordinate to Her Supreme Majesty, Queen Kelirehenna I, Lord of Sto Lat, Protector of the Eight Protectorates and Empress of the Long Thin Debated Piece Hubwards of Sto Kerrig.
[2] His daughter, however... we can only speculate. But inheriting the "touch(/slap) of Death" from the previously apprenticed father and the
de jure lineage from the mother (who was an adoptee) may have combined interestingly in what passes for near-Lamarkian heritability on the Disc. That and/or early-life exposure to Death's Domain, much as Melody Pond gained that 'little extra' through gestational exposure to the Time Vortex in Doctor Who.