Yes you can transfer your license to a different installation. You can just apply your existing license to this new server. Steps on how to do this are in Updating your FishEye license.
The Server IDs do not need to match up to be able to use this key elsewhere. That Server ID is only needed when you first creating a license key. Since it sounds like you already have a key available to you on https://my.atlassian.com you can just use that key on the new server.
Technically I believe the license agreement states something along the lines that you are only permitted to use this key on one installation at a time, however for the sake of migrations, you should be fine here.
If the old installation is going to be shutdown and not used, then you should nothing else to do. However if you still intend to use the old installation (and the new one at the same time) then you might need to obtain another license key. If this is just temporary, I'd recommend creating an evaluation key, which would last 30 days before expiring. If you need to use this for a longer period, you might need to purchase an additional license.
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