The Apex Legends mode that uses 3v3 team deathmatch over rounds is Arenas.
Arenas was introduced in Season 9 as a more focused alternative to the large battle-royale format. Two three-player squads fight on a compact map, with each round beginning after players buy weapons, healing items, ordnance, and ability charges using materials. Eliminated players return for the next round rather than waiting for a banner recovery.
The match uses a round-based scoring system. A team generally needs three round wins and a two-round lead, while a tied late game can reach a final sudden-death round. This structure rewards coordinated fights and resource management rather than looting across a huge map.
Mixtape is a playlist containing several rotating modes, Gun Run is a weapon-progression mode, and Control is an objective-based mode. Arenas was removed as a regular mode in Season 16, but that later availability change does not alter the historical identity described in the question. Apex has also brought back Arenas-style gameplay in later limited or returning formats.