My T5 (2018) drives ok. However I notice that when stopping at a crossing and pulling up hitting the pedal at once to the bottom does not give the expcted behaviour.
You would expect a "flying start" when doing this.
But the car drives away rather gently and the real revs/launch happens after 1-2 secs. Is this normal?

 

Not sure of the drive features in your car. My 2020 V90 T6 has economy, comfort and dynamic. Particularly in eco and comfort, when really stepping on the gas pedal from a standstill, I've noticed that the engine revs can get a little high while the trans gearing struggles to keep up. I've also noticed this, to a lesser degree, when traveling at speed, on the highway, and attempting to perform a quick pass around another car. Placing it in dynamic makes the car's standstill and at-speed/kickdown acceleration much more responsive IMO.

 

If you put it in sport and keep it in 2nd gear, I bet it will feel a lot different! 1st gear is good until about 10mph, so I am assuming its really giving it the beans in 2nd. I have a 2018 S60 T5. It might also be your traction control, is your car an awd?

 

It is a FWD so not a AWD. I only have a normal and ECO driving mede...
If traction control is operating I should see a light on the dash right?

 

I noticed that sometimes when pulling up with very ittle throttle you also feel a kind of "launch" after 1-2 secs when keeping the pedal just minor depressed.
Is this slack in the transmission or just the turbo awaking?

 

Cant speak much for the fwd, but traction control light wont show unless it is turned off. When my 06 S60s awd wasnt working, which in turn affect the anti-slip function. I could spin my tires real easy operating under FWD. 1st gear gets you started, after that its go time. Being a normal street car, the volvos probably werent built to be launched. I have a 2018 s60 awd and it moves pretty quick from a start. I am also not sure what kind. if any of torque limiting they did for the P3s