Taking character level 1 in Fighter gives Constitution saving throw proficiency vs. Ranger giving Dexterity saving throw proficiency. That's why I would suggest that Fighter is a better choice for a level dip. Two-weapon fighting works well with Bladesinger, which is another reason for that 1st level in Fighter to get Two-Weapon Fighting Style, which you cannot get until level 2 in Ranger.
The main benefit of the single level dip in Ranger is to avoid any reduction in access to higher level spell slots. That does make sense too. Depends exactly where your Bladesinger build is focused: if spell focused, then single-classed Wizard is best to access those highest level spells; mixed but still spell oriented, then a single level dip in Paladin or Ranger; if melee focused, then a one or two level dip in Fighter (Action Surge is so good!).
For a spell casting focused build, you should also consider Sorcerer. Taking 1st character level in Sorcerer gives Constitution saving throws for Concentration checks. At level 2 Sorcerer, you can quicken a spell once per long rest. That said, there's less motivation to level dip into Sorcerer in D&D 2024, as the Draconic archetype benefits are not given until Sorcerer level 3. In D&D 2024, those benefits were at Sorcerer level 1, making it a stronger level dip for the additional +1 HP/level and AC.