According to my conservative estimate, more guns were sold on Black Friday than there were marines (both active-duty and reserve) in 2017.
For the first question, I am unsure what the mismatch between the title and body of the source is. I do know that the FBI has released data on the number of background checks performed on Black Friday in 2017 (Friday, November 24). The FBI background check statistics for "Top 10 Highest Days/Weeks" are available here directly from the FBI. We can see two things.
- Between November 30, 1998 and April 30, 2019, the most firearm background checks performed on a single day was Black Friday, 2017 (November 24, 2017).
- In 2017, 203,086 firearm background checks were performed on Black Friday.
The quote in the question also includes "up from the previous single-day highs of 185,713 [firearm background checks] last year and 185,345 in 2015." These numbers can be also be verified directly from the linked webpage.
The Department of Defense budget for fiscal year 2017 is available online from the Department of Defense (emphasis added).
This budget funds an Army of 460,000 soldiers in the active Army, 335,000 soldiers in the Army National Guard, and 195,000 soldiers in the Army Reserve in FY 2017 – comprising 56 total Army brigade combat teams and associated enablers – and a Marine Corps of 182,000 active-duty Marines and 38,500 Marine reservists. For the Navy, the budget continues to grow the size, and, importantly, the capability, of the battle fleet; which grows from 280 ships today to 308 ships at the end of the FYDP and provides for 380,900 active-duty and reserve sailors in FY 2017. The budget also supports an Air Force of 491,700 active-duty, Reserve and National Guard Airmen, and includes 55 tactical fighter squadrons in FY 2017.
Thus, the 2017 budget allocated funding for and authorized a total of 220,500 marines (of which 182,000 are active-duty).
I couldn't find statistics for the number of guns sold for any year (including 2017). All statistics I could find were for background checks, so I assume this is how the industry keep track of gun sales.
To get a better idea of actual gun sales (which, as @femtoRgon points out "is not the same as the number of background checks"), I performed a rough, conservative estimate of the minimum number of guns sold on Black Friday 2017.
FBI data for background checks in 2017 is available here. Quoting two sentences (not contiguous within the document):
A mere humble beginning at that time considering in 2017, there were 25,235,215 firearm background checks processed.
103,985 [background check] denial decisions were provided in 2017.
Thus, 0.412% of background checks were denied in 2017. Using this percentage that applies for 2017 in general to specifically for Black Friday, 837 background checks were denied and 202,249 were passed. I couldn't find research on the ratio of gun sales to background checks; however, I believe a reasonable, conservative estimate is 1:1. Assuming that each background check resulted in a single gun sale, which is not the case as described in the linked article from the question, 202,249 guns were sold on Black Friday where background checks were performed. However, guns are also routinely sold without background checks having been performed.
This 2017 study is the most recent I could find on the subject. Miller et al. wrote:
22% (95% CI, 16% to 27%) of gun owners who reported obtaining their most recent firearm within the previous 2 years reported doing so without a background check.
Using a conservative estimate of 16% (the lower bound of the 95% confidence interval) and assuming the percentages apply to Black Friday 2017, 16% of guns sold that day were obtained without performing a background check. This means the 202,249 guns sold after performing background checks comprise 84% of gun sales on Black Friday and 240,772 guns total were sold on Black Friday 2017.
Were enough guns to arm the Marine Corps sold during Black Friday 2017?
Less background checks were performed than number of personnel approved by the Marine Corps budget. More background checks were performed than number of active-duty personnel approved by the Marine Corps budget. Using various assumptions allows calculation that around 20,000 more guns were sold than number of marines. Despite the use of an estimate, the theory that more guns were sold than there were active-duty marines is supported by the fact that an estimated 58,772 more guns were sold than there were active-duty marines. The claim is likely true.