No, the tar file format stores no information about the program that generated it.
Linux versions of the file utility can detect the difference between Solars and GNU tar generated files. It does this by extending the search to include the format and version information in the header. GNU generated tar files have
ustar (thats 2 spaces after ustar) whereas Solaris tar files have
You can make your Solaris version of file work in a similar manner to the linux one by adding the line
257 string ustar\040\040 USTAR tar archive (GNU)
above the line
257 string ustar USTAR tar archive
/etc/magic. Having said that I don't know and can't find a standard for encoding the version information so I would assume it's a hack that can't be relied on.
I can't say I've ever found a tar file that GNU tar couldn't handle that the Solaris tar could (the reverse is as you know not true) so in your situation I would just use
gtar to extract files from any tar file you have.
Regarding your question in the comments. That's not a great idea as an exit status >0 just means an error occurred which may be an error other than 'I don't understand this format'.