The true polymorph spell description states that the target's game statistics are replaced with those of the new form. When I true polymorph into a monster such as archmage, or shadow assassin, without a specific class or race, do I keep mine?
Humanoid NPCs/monsters usually have a tag appended to their humanoid creature type, which designates their race. An archmage is a humanoid, but its tag states that it can be any race. If you true polymorph into an archmage, you can keep your original race, or you can choose any other humanoid race (the caster chooses, if you didn't cast the spell yourself).
If you true polymorph into a shadow assassin (from Waterdeep: Dungeon of the Mad Mage), your creature type is undead, and your have no specified race.
In any case, you wouldn't carry over any PC racial traits that aren't specified in the new form's stats.
Normally only PCs have a class. Although the Dungeon Master's Guide provides a method for DMs to create NPCs with PC classes, there are no published monster/NPC stat blocks that include a PC class. If you true polymorph into an archmage or shadow assassin, you'll no longer have a PC class. You wouldn't carry over any PC class features that aren't specified in the new form's stats.
True Polymorph says:
The target's game statistics, including mental ability scores, are replaced by the statistics of the new form. It retains its alignment and personality.
What it means is "Put the character sheet aside and use the statblock of the new creature chosen". Literally everything is replaced. The original character is almost completely irrelevant to the polymorphed replacement, nothing carries over except for alignment and personality as indicated in the quote.
As far as race for humanoid creatures goes, that's covered in the Appendix B of the Monster Manual, under Customizing NPCs:
Racial Traits. You can add racial traits to an NPC. For example, a halfling druid might have a speed of 25 feet and the Lucky trait. Adding racial traits to an NPC doesn’t alter its challenge rating. For more on racial traits, see the Player’s Handbook.
If you want to modify an "Any Race" NPC to be a particular race, you consult the PHB and make the same adjustments you would to a player character.
Archmages can be of any race. The caster of true Polymorph would choose what race you become.
Medium humanoid (any race), any alignment
Armor Class 12 (15 with mage armor)
Hit Points 99 (18d8 + 18)
Speed 30 ft. (etc) (SRD, p. 395)
Since it is any race, the option is open to select one.
If the spell caster True Polymorphed your PC into a Drow, then the race is as specified in the stat block
Medium humanoid (elf), neutral evil
Armor Class 15 (chain shirt)
Hit Points 13 (3d8)
Speed 30 ft.