In the 90's Spider-Man animated series Venom bonds with Peter Parker/Spider-Man. After their separation and after bonding with Eddie Brock Venom still creates spider webbing. Since Venom is an alien and this has nothing to do with spiders how is he able to shoot webbing? The webbing isn't even a power, it's created by Spider-Man's webshooters.
The Venom symbiote (actually a member of the alien Klyntar race) bonded with Spider-Man on Battleworld (during the Secret Wars arc as stated in comments & other answers).
While the symbiote appeared to be producing 'webbing' for Spider-Man that he controlled by his thoughts, what it was actually doing was extending tendrils of its own mass and then disconnecting them from itself. The use of the symbiote in this way was purely because that is what Spider-Man wanted to do with it.
Once the symbiote was disconnected from Peter, it bonded with Eddy Brock, who being very familiar with Spider-Man, used the symbiote in the same way as a mode of transport to swing around New York (he also had access to the symbiotes memories of doing this for Spider-Man).
So the ability to shoot webs (or appear to), is not actually a power gained from bonding with Spider-Man, but just an expression of the symbiote's inate ability of manipulating and disconnecting it's own mass.
This ability has been used by different symbiote/host combinations in different ways, e.g. the Carnage/Cletus Cassidy pairing would throw spears, knives, axes, etc. made up of the symbiote's mass.
Once disconnected from the symbiote's main mass, these extensions will disintegrate after a period of time, much like Spider-Man's webbing, but not because it was something 'learned' from Spider-Man.
When he first acquired the "suit" (as he initially thought it) it produced webbing for him without any need for his web-shooters.
He presumed it was some sort of high tech alien fabric that molded to his form & the form he wanted through some sort of inbuilt empathy tech & the production of webbing was part of that.
I recall that in the original comic version it appeared in he was involved in a super-hero / super-villain mass-battle orchestrated by one of the mega powerful types in the marvel universe, they'd been kidnapped to an alien planet to fight in a battle-royal. As mentioned by FuzzyBoots in comments this was the Secret Wars arc.
But the long & short is the suit symbiote produced webbing for him without his web-guns.
Something that as it eventually turned out was the venom symbiote forming its own substance to fulfill the desires of its host to the best of its ability.
So having learned to do it for Pete of course it could repeat the trick for any new host.
https://en.wikipedia.org/wiki/Symbiote_(comics) things have moved on somewhat more than apace since I last read any of these comics it seems, back then it was just Venom & Carnage, now they're all over the shop in their thousands it seems.
Like the Xenomorph in the film "Alien" the symbiote takes on desirable genetic characteristics. Second the Symbiotes fibrous nature allows it to extend portions of itself, that's it's webbing. The symbiote also possesses distinctive memories for a time it spent with Parker to adapt a means of producing "Mimicry webs"
It may be conjecture, but the symbiote's ability to "enhance the genes" of its host may be responsible for this. all Klntar symbiotes are said to "take the best parts of former hosts into themselves", and we know specifically the Venom symbiote maintained both its look, and many of Peter Parker's enhanced abilities (strength, speed, agility, wall crawling) that it likely got from connecting to his spider-enhanced DNA. As you are speaking specifically of the animated series, we know Peter never "produced his own webbing", but that the spider bite (in his own words) passed on an "instinctive knowledge of how to combine certain enzymes to produce this sticky filament." He said so right in the series himself...
In the same cartoon series, we saw Peter, when he was under the effects of his "mutation disease" be able to produce organic webbing. That wouldn't be possible if he didn't have the genes for it. In fact, a few different Animated series have shown Man spider being able to make his own webs
Possible answer: Outside of being tendrils, as the first time they meet we hear Eddie Brock/Venom tell Spidey "tsk tsk, such low quality webbing. Here! try some of _ours!", the symbiote likely took sample of Peter's "spider genes" and enhanced them, allowing him to produce bio-organic webbing similar to the kind he can make as the "Man spider". As to what the suit uses as the raw material to make this webbing, I can't be sure, but for the limited time Peter from earth 616 had organic webbing, he said "Eat a lot of starchy food, and you'll be fine."
Symbiotes adapt to their host's powers, and since Peter was the suit's first host, it stayed like him even after it bonded with Eddie. (Hence the spider symbol still being on Venom's chest.)
Since Spider-Man had the suit sling 'webs', it continues to do that to this day even without Peter in the suit.