Tech doesn't usually measure COGS, because the cost of another user on a website is near zero.
At the big tech (household name) I work we definitely spend a lot of energy talking about COGS a engineers. We have a LOT of data, and we spend a lot of money on cloud services to keep our product running - you want to add a new piece of data to this database to support some new feature? Go calculate how much it'll cost in storage and compute and if it's above some threshold, request special approval. You want to improve performance by caching our indexing? Let's talk about how much that'll be in COGS. This stuff ain't free, all those users add up.
The interesting bit is that all this COGS isn't actually money that transfers hands, the cloud we use is our own. But it isn't free and internal accounting is taken seriously.
I think "COGS" as discussed here would probably be a rollup of both? Not sure; while I've been in orgs where the cost center/profit center distinction was made, I haven't run into a situation where COGS was a metric I saw used.
Customer Acquisition Cost gets used in subscription businesses to measure sales and marketing performance. It can be relevant alongside COGS if you're manufacturing the thing you're selling subscriptions to, but that's kinda rare.