Is a 1099-MISC the same as a "government (Form 1099-G)"?

No, they are not the same at all. 1099-MISC is not issued by the government.

 A 1099-MISC is for any other miscellaneous income you received. Here are the other types of income that will be reported on Form 1099-MISC:

• $10 or more in royalties or broker payments in lieu of dividends or tax-exempt interest.
• $600 or more in:
  • Rents.
  • Prizes and awards.
  • Other income payments.
  • Medical and health care payments.
  • Crop insurance proceeds.
  • Cash payments for fish (or other aquatic life) you purchase from anyone engaged in the trade or business of catching fish.
  • Generally, the cash paid from a notional principal contract to an individual, partnership, or estate.
  • Payments to an attorney.
  • Any fishing boat proceeds.
  • $5,000 or more of direct sales of consumer products for resale