Shreyas Iyer was the third Indian player and 40th overall to make two 50+ scores on debut:

Earlier Dilawar Hussain (59 & 57 in 1934) and SM Gavaskar (65 and 67* in 1971) had achieved this.
From the above table, we can also see that Shreyas was the first Indian player and 16th overall to score a century and a fifty+ on debut. This 16 included the special cases of LG Rowe and Yasir Hameed who scored twin centuries on debut.
CG Greenidge (93 & 107 in 1974) and AN Cook (104 & 60* in 2006) did so against India.
Now we look at the most runs made in debut Test (this is not the same as the above chart, as some would not have made 50+ in both innings.
Here the cutoff is 160 runs.

Iyer made 105 + 65 =170.
This is the third highest for India (after S Dhawan’s 187 and RG Sharma’s 177) and 28th overall. J Neesham also made 170 on debut, but is ranked above Iyer as his performance was earlier.
Also note CG Greenidge’s 200 (93 and 107) J Neesham’s 170 (33 & 137*), MJ Clarke’s 168 (151+ 17), AN Cook’s 164 (60 & 104*), DL Houghton’s 162 (121 & 41*) and CH Lloyd’s 160 (82 & 78*) against India
Another curiosity is that of the forgotten BM Laird’s 167 runs, which is the highest match total on debut by someone who did not score a century. Next are CH Lloyd and Fakhar Zaman both with 160.
And finally:
New Zealand was set to score 284 in the 4th innings to win this Test.
Guess what the highest winning chase at this venue was?

This doesn’t mean much as the Green Park has had many drawn matches, and is probably the most batting-friendly pitch among India’s older Test centres