You can create as many signed URLs as you wish. Depending on your requirement, however, there is a practical limitation on the number of unique presigned URLs for the exact same object.
S3 (in S3 regions that were first deployed before 2014) supports two authentication algorithms, V2 and V4, and the signed urls look very different since the algorithms are very different.
In V2, the signed URL for a given expiration time will always look the same, if signed by the same AWS key.V4 is different.
Note that both forms of signed URL are tamper-resistant -- the expiration time is embedded in the url, but attempting to tweak it after signing will invalidate the signing and make it useless.
Hope this helped!