What is the default Same-Sex Marriage game rule in CK3?

Default: Same-sex marriage is not available. (patch 1.20.1.2)

Does changing the Same-Sex Marriage rule disable achievements?

Yes for “Accepted”: the file gives it the flag blocks_achievements. The other settings keep achievements on.

Game rule · Game Modes

Same-Sex Marriage

Default
Accepted ✕

Same-Sex Marriage is a game rule in Crusader Kings III (Game Modes) with 2 settings: Default (the default) and Accepted. “Accepted” disables achievements. Patch 1.20.1.2, from game/common/game_rules/00_game_rules.txt.

This section as JSON: /raw/game-rules.json

Quick answers (2)

What is the default Same-Sex Marriage game rule in CK3?: Default: Same-sex marriage is not available. (patch 1.20.1.2)
Does changing the Same-Sex Marriage rule disable achievements?: Yes for “Accepted”: the file gives it the flag blocks_achievements. The other settings keep achievements on.

2 settings

Default
Default
Same-sex marriage is not available.
default_same_sex_marriage
Accepted
Disables achievements
Same-sex marriage and concubinage is available for everyone whose Faith accepts it, and whose culture practices concubinage. To enable for everyone set the Same-Sex Relations Game Rule to Accepted!
The script checks it once, in 1 place
Scripted triggers 1
- allowed_to_marry_same_sex_trigger
Every has_game_rule = accepted_same_sex_marriage in game/common and game/events, by the top-level object it sits in.
accepted_same_sex_marriage

Source: game/common/game_rules/00_game_rules.txt line 320 (key same_sex_marriage), modifiers from game/common/modifiers, names from the English localisation; Crusader Kings III 1.20.1.2.