A second tunnel is not needed. Some sums:
There are 4 paths for passenger trains every hour (departures from St. Pancras at :01, :04, :31 and :34) and a small number of hours with a 5th extra path departing St Pancras at :16
currently only eurostar use these paths, but theoretically any other operator could use them if they wanted to run services to the UK in the future.
So, using only the current paths and existing e320 trains, the channel tunnel has a capacity of 4 x 900 = 3600 passengers per hour, rising to 4500 passengers for the hours with the 5th path
If you switched to using duplex trains, with a capacity of ~1350 passengers per train, this goes up to 5400 passengers per hour (6750 for hours with a 5th path)
My understanding is, currently St. Pancras has a capacity of ~1500 passengers per hour (source:
https://twitter.com/EurostarJustinp/status/1574711574337495041/photo/1)
There are plans to increase St. Pancras capacity, and of course some of the passengers could be coming from Ebbsfleet and Ashford, but I doubt the capacity increase will come even close to the theoretical limit of 6750 passengers per hour before you start needing a second tunnel!
Though I would hope the capacity upgrade brings us a lot closer to the 3600 passengers maximum capacity that can be achieved with current rolling stock, because it would allow eurostar (and potential on-rail competitors, if they ever show up) to grow, and take a larger share of the London to Paris/Brussels/Amsterdam air market