Do I need to hurry in Cyberpunk 2077?

The nature of main quest-line in Cyberpunk 2077 compels you to finish it as soon as possible and ignore various side-quests.

However, when playing open world RPG, such approach will generally make you miss big parts of content.

Is there actually penalty in main-quest for taking too much time? Or is the pressure only story device without any real effect?

No, you generally don't need to hurry. Most quest states will wait for you until you return to play them. So you're free to do other things in the game and let those quests wait.

This also applies to side quests which you can accept, often via phone, by saying stuff like "yes I will come over to deal with it" (or similar), but you don't actually need to do that right away.

There is only one exception, though. It is the quest "Happy Together" where you need to talk to your neighbor (the one who doesn't want to speak with his police coworkers).

The quest requests you to talk to Barry after he blocked to talk to his coworkers. You can't speak right away with him, but shouldn't wait to long either (skipping a few hours is enough). If you take to much time, then the quest can fail. You can read more about that quest here: https://www.thegamer.com/cyberpunk-2077-happy-together-save-barry/