Your Mac clipboard holds one thing. Copy something else and the first one is gone, with no warning and no undo. That is why "how to copy multiple things at once on a Mac" gets searched so often, and why most of the answers are unsatisfying: they answer a different question than the one being asked.
Two questions hide behind one search
The first is about files. You select six PDFs in Finder, copy them, and paste them into another folder. That works today and always has.
The second is about everything else. You have a tracking number in one tab, an address in another, and a reference code in an email, and you want all three before you start filling in the form. macOS has no answer for that on its own. You go back and forth instead, one copy at a time, which is exactly the thing you were trying to avoid.
Worth separating the two, because only one of them needs extra software.
Copying several files at once
In Finder, click the first file, then shift-click the last one to take everything in between. To pick out files that are not next to each other, hold ⌘ and click each one. Press ⌘C, switch to the destination folder, and press ⌘V. All of them arrive.
There is a detail here that trips people up later. Your clipboard now holds a single item that points at several files. It is not six clipboard entries. So the moment you copy anything else, the entire group is gone together, and there is no way to get it back except reselecting the files.
Same goes for ⌥⌘C, which copies the file paths as text rather than the files themselves. Useful when you need the paths for a script. Still one slot.
Copying several separate things
This is the one with no built-in fix. One slot, and ⌘C overwrites it. Universal Clipboard stretches that slot across your iPhone and iPad, but it is still a single slot, just a longer one.
So people improvise. A scratch note in TextEdit, everything pasted into it, then copied back out one line at a time. I did this for years and it works right up until you have four near-identical order numbers in a note and no idea which is which. The workaround costs more attention than the original problem.
Clipboard history is the actual answer
A clipboard manager sits in the background and keeps a copy of everything you copy. The single slot still exists, because that is how macOS works, but nothing falls out of it permanently. Copy ten things in a row, then go back and paste any of them in any order.
That changes the shape of the work. Instead of alternating between two windows ten times, you copy everything from the first window, switch once, and paste your way down the list.
What this looks like in reClip
Press ⇧⌘V and a panel opens along the bottom of the screen with your recent clips as cards. Press 1 through 9 to paste one directly, or use the arrow keys and Enter. The panel closes on its own afterwards, so it stays out of the way.
Copying several files makes one clip, the way macOS does it, but the card tells you what is in it: a preview of the first file, then the other names listed underneath with a count. A six-file copy reads as six files instead of looking identical to a one-file copy. Images and files are captured on the free tier as of version 1.4.2, so this is not something you have to pay to see.
The free version keeps your last 20 clips. That is enough for the copy-several-things-then-paste-them problem, which usually plays out over a few minutes. Pro removes the limit and adds search, which matters when you want something you copied on Tuesday.
A workflow that survives real work
Filling in a form from three sources. Open all three, copy from each one without stopping to paste, then go to the form and press ⇧⌘V. The three clips are the top three cards. Press 3, 2, 1 as you move down the fields.
Moving files out of a download folder. Select what you want, ⌘C, paste into the destination. If you then copy something else and realise you were not finished, the earlier file clip is still in the panel with its names on it, so you can see exactly which files you had picked.
Something I still do by hand, and probably should not: pasting a batch of hex codes into a stylesheet one at a time. Groups would fix it. I keep not setting it up.
What about the iPhone
Universal Clipboard already moves a single copy between Apple devices, with the usual caveats about both devices being awake and on the same network. reClip syncs your history through your own iCloud account instead, so what you copied on the Mac is on the phone later, not just in the next thirty seconds. Nothing goes through our servers, because there is no server in the path. Syncing clipboard between Mac and iPhone goes into how that works.
The short version
Several files at once: Finder does it, ⌘-click to select, ⌘C to copy, and remember it is one clipboard item that disappears as a unit. Several unrelated things at once: macOS will not do it, and no shortcut will make it. You need something recording your copies.
reClip is free for the first 20 clips and needs macOS 15.5 or later. Download it and copy four things in a row to see whether the difference is worth anything to you. It took me about a day to stop keeping a scratch note open.