According to the above pages, NM-2E2W is supported on 3620 platform for all IOS version and NM-2FE2W is supported on 3620 with ios version 12.0 or later.
I'm not clear from your description about what the exact module you have. If it is NM-2E2W and you can't see it on your 3620, either the cisco documentation is un-reliable (too bad) or you have a defective hardware (really too bad)
We are pleased to announce availability of Beta software for 16.6.3. 16.6.3 will be the second rebuild on the 16.6 release train targeted towards Catalyst 9500/9400/9300/3850/3650 switching platforms. We are looking for early feedback from custome...