Yes, some utility depending on the game.

Back in the arcades, an inactive game switched to “attract mode”. Basically it would present pre-recorded fun segments of the game or movies while waiting. Pressing a button or dropping in money would end the attract loop.

These days the purposes can be different. One is to identify the main controller. On a computer, do they touch a keyboard/mouse, or do they touch the gamepad? If they touch the keyboard/mouse they get the labels for that, “Press Enter", “Press Space”, etc. If they touch a gamepad they might get press “A”, or press button “1” (depending on the labeling). Also if they touch a gamepad and there are multiple attached, which one? Whatever they touch gets the control. They also provide an opportunity to adjust volume levels or contrast or other settings to be comfortable before the game starts.

It can serve as a screen-saver of sorts, keeping you from seeing a static screen. First-party certification forbids displaying a static screen for too long for a bunch of reasons, including screen burn-in but also to show the game hasn't crashed. You have to display animation anyway, might as well make it something interesting.

Plus they also keep the original functionality of an attract mode when the game is shown at a game store or taken to a trade show. Some games do that better than others. The bad ones just say “Press Start” or similar. The good ones show interesting tidbits just like the old arcade games.

Basically, if you put anything interesting before the main menu, then the player might want to continue looking at that instead of proceeding immediately to the main menu. So you give the player the option of looking at it, and pressing a key to go to the menu.

As for why you would put something before the main menu, well, there are lots of reasons. Maybe you need time to load the menu, so you put up a loading screen. Maybe you have a beautiful title image that you don't want to immediately cover up with a menu. Maybe you want to pay homage to the attract modes of arcade machines.

Ultimately it's a stylistic choice. There's nothing wrong with going straight to the menu, it's just that various developers have chosen not to do so, for various reasons.

So you can listen to the theme tune ?

I think it is a quite clever way for players to see if their input (controller/mouse/keyboard) is working. If you just get into the main menu or game right away some people (mainly children or people with mental disabilities) might think “the menu/game is not working", but if it literally says “Press any key” some of those people might* make the connection that it is the input device that is the issue.

I doubt this is the reason it was added in the first place but I think it is a nice side-effect of having it.

* I'm just speculating and haven't seen any research on this but I'm certain small UI changes can have a big impact for usability.

One reason could be that the console manufacturers require it for certification ? As long as that is true, developers have to do it, even if they don't need it, or know why it's useful ?

And the answer to “why do they require it for certification” could be any of the other answers given in this thread. But for example, you could detect the primary controller in the main menu, you don't necessarily need a separate splash screen for that.

I guess having all games on the platform follow the same pattern, at least in the first screen, is something that is desirable for the platform owner.

I always prefered games which you can skip the intro video and credits.