Do Mage Armor and Bladesinger stack?

features that changes your BASE AC are mutually exclusive.

Unarmored base ac is 10

Studded leather base ac is 12

Breastplate base ac 14

Full plate base ac 18

If you wear studded leather and breastplate, you base ac is just base for breastplate, 14. Different sources of BASE AC are mutually exclusive. You just pick one.

Relative armore bonuses can usually add to any base ac.

Dexterity modifier can be added to most base ac values, but different armor may have different rules on dexmod. Light armor can add any dexmod. Med armor can add up to +2. Plate armor is unaffected by dexmod.

A shield is +2. And you could add that +2 to studded leather (base 12) or breastplate (base 14) or fullplate (base 18).

The Shield spell adds a relative +5 to your ac for a short period, whether you are unarmored or wearinf plate or in between.

Mage Armor says: "target’s base AC becomes 13 plus its Dexterity modifier. The spell ends early if the target dons armor."

So if you cast mage armor, your base ac is 13 and you can add dexmod. Spell says it ends if you wear armor, but a shield isnt armor so you could cast mage armor, get a base ac of 13+dexmod, and wield a shield for an extra +2 to your ac.

Bladesong in Tashas says you gain a relative bonus to your ac of +intmod. But it does add several extra caveats, namely it doesnt work if you are wearing medium or heavy armor or using a shield.

So that leaves light armor for a base ac of 12 (studded) plus you dexmod (maybe a +3 or +4) and blade song gives you an extra +intmod to your ac.

So mage armor is a base ac of 13+dexmod, and you cant wear any armor for it to work. And bladesong says it gives you a +intmod to your ac and it doesnt work with med/heavy armor or shield.

So they will work together, giving you an ac of 13 + dexmod +intmod, as long as you dont wear any other armor and dont weild a shield.

Alternatively, Instead of blade singing, take 3 levels of artificer and the armorer subclass. You get heavy plate armor that ignores strength requirement and 3 weapons attached to your armor that use INT for attack and damage. Dump str and dex, max your int and con, and then go all wizard.

Wizard in my opinion is best made for kiting. Move fast, stay out of melee range, and use true strike for a bow attack, using int for attack and damage roles. And when the need or opportunity arrives, use a big spell for aoe or big battlefield control effects. Mage armor and Shield spell, and try to stay in the back.