So most fungi do have a lifespan, they have teleomere decay, and when you're cloning mushrooms (from propagating mycelia) you have to let them go to fruit (the part that looks like a mushroom) every now and then. It's a pain in the ass.

But like the other poster said, they play it fast and loose with which part you consider the "organism". My favorite thing is that they do cytosolic streaming. Genetics can be a pain on mushrooms because not only do they share nutrients and metabolic burden through mycelia, they can share nuclei.

One of the weird convienent realities we used extensively is that cells are big enough you can spread them over a petri dish with a little loop, and if you diluted the initial sample enough, the colonies that developed were, practically speaking, from one parent cell. So you could try to modify a bunch, and then plate them (spreading the cells around) and pick individual colonies that were all clones from a single parent. Fungi mycelia means the nucleus isn't stuck in one cell. It also means expression levels can be variable (some cells will have multiple nuclei, and then later maybe they don't).

Fungi are a godamn pain in the ass to study. They're not mysterious, they're not alien, they're just fucking assholes.

Based on the Wikipedia article on biological immortality referencing species that live for a couple hundred years and the Wikipedia page on armillaria ostoyae mentioning living specimens that are multiple millenia old (and thousands of acres large!), I'm guessing that may be what the prof is referring to?

I'm guessing they're talking about clones?

Yeah, lots of living beings do that.