I thank you JustMe'D for your prompt & welcome reply.
To answer your question accurately. "No I didn't see an option to Save", but then again I only thought to Long-Press on the attachment(not Tap) & when I did select to Long-Press(while in gMail), only 3 options were presented. The first of which was 'Copy'(which I'd think would be the correct avenue). However, after selecting 'Copy', I was unable to 'Paste' thee attachment anywhere within my iPod Touch 5.
The second option was to 'Save the .Mov attachment to a "Read Later List"(which as far as I can See/Find does nothing). Besides the file attachment I'm trying to save(for offline playback) is a .Mov file, not a Word/Text document ! Then there's a third option, which I can't recall rite now & I don't have the iPod in front of me so as to check, however, one thing I'm 100 % sure of, is "it isn't an option to Save". In closing allow me to add that I fully appreciate your Screen-Shots, I only wish "I" knew how to add SS's of what I'm seeing while attempting to save thee attachment onto my stock iPod Touch 5G(iOs 7) while in gMail.
PS:
I can't help but wonder if Verizon's proprietary software(which my stock iPod Touch 5G 'iOs 7' doesn't have) is what's allowing you to save video attachments to your iPhone.