I was initially confused by vin's comment too. But I suspect that it reflects an in vivo system where ancillary signaling is altered through some transcription / epigenetic mechanism. So measuring the protein's activity in an in vivo assay through a surrogate (ability to phosphorylate its target protein X after agonist treatment) may end up being different, but it would be more a reflection that the signaling cascade is altered, not that the pure activity of the protein has been altered.
The other possibility is that since some proteins require post-translational modification (for example, phosphorylation) during translation in order to achieve their structure, perhaps codon usage / tRNA availability could impact this pathway. But again, this doesn't strictly apply to the example you have in mind. It would just reflect altered processing and the cause for the change in activity would not be the amino acid with the altered codon, but rather a change in the percent of protein appropriately processed.
I don't think vin intended to suggest that a pure protein generated from variable codon usage would have different function... but like you, that's how I first understood it.
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