The light means that the ECU is getting a signal from somewhere that's "out of range". It could be something was previously damaged, or the engine work could have disturbed something in one of the electrical connections, or at the outside, the new motor and/or any sensors attached to it are slightly incompatible with the rest of the car.
How did your old engine die? If it suffered a catastrophic failure and it dumped a load of oil through the exhaust (e.g. if it dropped a valve and punctured a piston.. loads of smoke out of the back) then the lambda sensor(s) and Cat' could have been contaminated. If the engine just seized or broke a rod etc. then the exhaust system would be oblivious to all that, and a contaminated Cat/lambda would be very unlikely.
What actual code is the ECU reporting? I bet someone on here had it and fixed it.
I don't know how the ECU would know the Cat' is damaged.. all it's got to go on is the lambda sensor. Your car may have a pre-Cat and also a post-Cat lambda(s)... so the post-cat sensor might be reporting something odd. Give it an inspection. But if you have just one sensor, it will be fitted pre-Cat.. so there's nothing to tell the ECU what's happening downstream of that. Your Cat is likely okay.. even unburnt oil tends to not kill them.
Check that the lambda *connector* is immaculate, clean and tight. It may be worth fitting another new one, but buy an NTK or Bosch doodah, not a spliced-in cheapo' one. Lambdao's things work on resistance variances of less than half an Amp or something... so you need one with an integrated plug fitted, not one that you have to cut and splice-in to your original plug.
It won't be your engine. That starts and runs okay, apparently... so it's fine. You won't do any damage to it driving with the light on. At worst, it will use more fuel and be down on power because the ECU doesn't know what the lambda (or something) is telling it, so goes "cautious" (i.e. rich).. since this doesn't damage the engine. How is your fuel economy/power? I guess "no difference"?
If you accelerate slooooowly in the same gear from 1000rpm to 2000rpm.. have a "seat of your pants" feel for what the engine is doing. Do you get a hesitation around the 1500-1800 rpm mark? The lambda includes a heater to keep it warm at low revs, and that turns off when the engine is using more revs. They turn off around 1600rpm ... so if your lambda is a bit weedy, you'll notice a hesitation when the heater cuts out and leaves the sensor to fend for itself.
Ralf S.