Yes, but the surge would proceed the barometric lowest point which is the eye. The surge comes from the rotating wind blowing the water.
The worst storm surge I have seen was some 3m rise by typhoon Hato - making landfall some 300 km south of us! That came on top of an astonomical high; the maximum of the storm and the maximum monthly high were within a few hours of one another. Major flooding all over the city. At our neighbours in Macau it was worse, some 4-5m surge. That was still quite a way from the eye.
A year later Mangkut, also making landfall a couple hundred km south, gave a near 2m storm surge. Luckily no extreme high at that moment but still serious flooding.
So really, you don't have to be in the eye of the storm to see serious surges. The direction of the winds and the regular tides are at least as much of a factor. That you see several meters of surge is more than can be explained by just the pressure, pressure alone gives maybe a 1m surge, the rest is winds pushing the water towards the land. If you have the winds blow the opposite way, there will be no storm surge at all even with very low pressure.
In these cases the Observatory did warn for storm surges but not to what extent. That was clear only after the storm passed. That I remember the names should tell you something about the severity of the storms. I don't even remember the name of the typhoon that gave us a T8 warning signal about a month ago. We just enjoyed the day off.
To monitor such surges, your tidal gauge has to be mounted very securely, in a place where the waves don't break it. Storms are really destructive - after Mangkut several beaches lost all their trees, as in trees were completely gone, not just blown over.