Lets look at levels 8 through 12. This is probably the sweet spot for 5e. Lets assume you are taking feats rather than ability score bumps. Your stats will be as such.
AC 19 (Mage Armor + Dex + Int) with the possibility of a 24 including shield. that's better than most fighters and paladins with a sword and board. This is without concentration spells like blur. You can still use Mirror Image. This means you can use your concentration for attack spells.
Attacks (Assume Rapier) - 1d8+3 x2. Most sword and board warriors will be doing 1d8+7 with no bonus action attack
Bonus action attack - Two weapon fighting warrior does 1d8+3. You have spells like Flaming sphere (2d6 and can be upgraded), Melf's minute Meteor (4d6 per round).
I think the Bladesinger should be making use of his multi-attack. I think spells like GRB and Booming blade benefit the Eldritch Knight more than the Bladesinger.
AC 19 (Mage Armor + Dex + Int) with the possibility of a 24 including shield. that's better than most fighters and paladins with a sword and board. This is without concentration spells like blur. You can still use Mirror Image. This means you can use your concentration for attack spells.
Attacks (Assume Rapier) - 1d8+3 x2. Most sword and board warriors will be doing 1d8+7 with no bonus action attack
Bonus action attack - Two weapon fighting warrior does 1d8+3. You have spells like Flaming sphere (2d6 and can be upgraded), Melf's minute Meteor (4d6 per round).
I think the Bladesinger should be making use of his multi-attack. I think spells like GRB and Booming blade benefit the Eldritch Knight more than the Bladesinger.