Glad it got solved for you!
You can read more about this function here: https://helpx.adobe.com/si/illustrator/using/moving-aligning-distributing-objects.html (search for 'Paste Remembers Layers'). Here is the excerpt:
By default, Paste Remembers Layers is off, and artwork is pasted into whichever layer is active in the Layers panel. When Paste Remembers Layers is on, artwork is pasted into the layer from which it was copied, regardless of which layer is active in the Layers panel.

So I am closing this as Resolved then, since isolating a layer now is not required :)
However, a hotkey for this command is something we need still... but this is another story.

It also solves the problem with pasting from outside AI - when I use Paste in place now, the picture from outside is placed in the selected layer. JIPPI! :-)

BRAVO! IT WORKS!! Egor, you solved it! Thank you, thank you, thank you!! I have never looked at that function, because I never understood what it meant...it's not always easy to use AI in english, because it's not my first language, and some of the expressions seems quite cryptical. But you really made my day, and pasting in the future will be much easier. :-)

Kristin, can you check please if you have 'Paste Remembers Layers' enabled in the Layers panel’s flyout menu?
If yes — does it change the behavior with Paste in Place if you disable it?

So I found a report about this, with 4 votes at the moment, and added some details for it to get more relevant in search results for those who can try to report something like this again:
http://illustrator.uservoice.com/forums/601447/suggestions/36922342
Please upvote it!

Hello again, Egor! My version is 27.7 (64 bit). And if you fix the 20% bug, I will not need the "Isolate hotkey" so much, I think. But sometimes it's very useful to isolate the layer. I design medals, and use a lot of layers/groups in one design. Sometimes it can be difficult to isolate obects because there are so many layers/objects above, and then it\s easier to select the layer and isolate. I also use a lot of masks, and then it can be difficult to select the right object to isolate by dobbel-click. But if it's very difficult to make, it's not essential. I can manage without... :-)

Egor, please don't misunderstand my somewhat harsh comments, I love the Adobe products, and are really dependent on them on daily basis. I really appreciate the continuously development with a lot of new, very intelligent features. But the apps are really big, complicated animals, sometimes with their own will, and can be quite annoying and frustrating when they don't behave... :-)

Egor, that's correct - when I use paste in place, even if I select another layer, the object will be pasted in the original layer, above the original shape.. I think it changed some years ago with a new version, and I remember thinking this was rather stupid, changing a well-functioning command/short-cut. But I have been in the business since 1996, and have experienced a lot of stupid new ideas and upgrades, especially changes of nice short-cuts I have been using for years. So I didn't think of this change as a bug. But I'm using a PC, not a Mac, could this have anything to do with it? I have experienced Adobe doesn't always care so much about us PC/Android-people, as you care about Apple-users...? ;-) But I'm very grateful for your interest in this problem, Egor - thank you very much :-) I hoper you figure it out, in one way or another.

Kristin, this is strange.
Let me ask this, just to be sure:
when you copy the objects on Layer 1, then select the Layer 2 and use Paste in Place command — do you get your objects pasted into Layer 1 still, despite you have Layer 2 highlighted in Layers?
This is NOT how Ai behaves for me (I checked three latest versions). I know Paste in Front behaves like this, but not Paste in place...

Enter Isolation Mode command (which indeed can’t be assigned to a hotkey and works with the selection IN Layers only) is a workaround for this, but I think the problem with Paste in Place is just a bug :/
Let me test it more.

Hello Egor, and thank you for answering me! But it doesn't solve my request. I made a short-cut the way you describe, but it works only on objects, not on layers. Some years ago "Paste in place" worked like this: I chose the layer I wanted to paste into, used the shift-ctrl-v-command, and the object were placed where I wanted it.
Today, when I paste in place, the object is placed in the original layer, above the original objects. To get the new objects to an new layer, I have to either drag them into the new layer (I use a lot of different layers in my work, and it's quite annoying), or I use the isolate layer command. But the short-cut you made me make, doesn't work on layers, only objects - so I still need my Isolate layer-short-cut/button :-)

Kristin, Illustrator allows to assign a hotkey to this command already.
Open the Keyboard Shortcuts dialog (Ctrl + Alt + Shift + K).
Switch to Menu Command from Tools in the dropdown list.
In the search field type 'isolate' — Illustrator will filter the list of available commands below to just one position, Isolate Selected Object in the Other Object section (or you can just scroll the list to the very bottom).
Assign the hotkey you need (I am used to having it on Ctrl+>).

Also you might want to vote for having a hotkey for Isolate One Level Back — which Ai does not allow to have a hotkey for: https://illustrator.uservoice.com/forums/333657-illustrator-feature-requests/suggestions/32403778-make-a-hotkey-to-go-one-level-up-in-isolation-mode

But I want to know a bit more about the behavior you talk about, the pasting at the top of the level.
I know about this, and the isolation trick is something I use too to mitigate this when I need to create one or more new shapes directly above a certain object. But you say there was a time Ai used to do it differently? Like draw new stuff directly above the selected object? Like the way Affinity Designer does now?

Because there is a request to have a 'Draw above / in front' method: https://illustrator.uservoice.com/forums/333657-illustrator-desktop-feature-requests/suggestions/45005431--draw-in-front-button
...and I don’t recall Ai having it like this ever — just rechecked older versions.

But I think you refer to pasting specifically. I rechecked this too, in the latest version — Ai still pastes directly above the selected object, if you use Paste in Front command, just like it used to do. Can you recheck this too?