Multiple concentration is (almost) always overpowered
Clarification and Assumptions
My answer is based on the following clarification of the proposed change:
You may choose to cast a spell with a casting time of 1 Action or 1 Bonus Action when starting Bladesong. If the spell has a casting time of 1 Action, you must use your Action to cast the spell on the same turn in which you start your Bladesong. If the spell has a casting time of 1 Bonus Action, you may cast it as part of the Bonus Action used to start your Bladesong.
The chosen spell must only target yourself and have a range of self or touch and require concentration, however you maintain the effect via your continued Bladesong instead of concentrating.
Also, I am writing this under the assumption you laid out in your question:
I'm planning to continue playing melee style with a 1d6 hp pool and not intentionally abuse it for some ranged double-conc with something like minute meteors/watery sphere.
If this assumption is not followed, multiple concentration is even more overpowered.
What spells can you use together?
Given your desire for a melee playstyle, the "restriction" of Range: Self or Touch does not change your choices very much. Most of the good options already meet those requirements. Without multiple concentration, you are usually forced to pick between offense and defense. Now, you can pick any two of the following spells:
- Shadow blade grants you more damage that a greatsword, while still being finesse and thrown.
- Blur gives most enemies disadvantage against you, which may be comparable to adding INT to AC, depending on the situation.
- Greater invisibility is an upgrade from blur that also raises your offensive capabilities.
- Fly allows you to avoid most of the damage from melee enemies.
- Protection from Evil and Good makes you very hard for a wide array of opponents to kill you.
Combinations like blur and shadow blade are already pretty awesome; in reality, you could combine one of the above spells with any other wizard spell (restricted by your agreement to "not intentionally abuse it").
So is it imbalanced?
In order to achieve something like multiple concentration, you would normally need two PCs or the 7th-level spell simulacrum. That alone should set off warning bells.
As the appropriately-named Quadratic Wizard points out in his answer to "What are the impacts of permitting casters to concentrate on 2 spells?", the DMG (p. 263) also believes multiple concentration to be a bad idea:
Beware of adding anything to your game that allows a character to concentrate on more than one effect at a time, use more than one reaction or bonus action per round, or attune to more than three magic items at a time. Rules and game elements that override the rules for concentration, reactions, bonus actions, and magic item attunement can seriously unbalance or overcomplicate your game.
I believe that most of these problems could be solved by the DM diversifying their encounters and playing the enemies more realistically.
He should try hitting you in the NADs (non-AC defenses). A simple 1st-level entangle spell could lock you down for the whole fight. Good luck making a Strength save/check with a -2. Even a fireball is reasonable. You may have decent Dexterity, but the fighter has way more HP than you. You might also have absorb elements, but you can't cast that and shield.
Also, there is no such thing as aggro in 5e. Reasonably-intelligent foes should rapidly give up on attacking you. Once they realize that your attacks won't pose a significant threat, some opponents should just eat the opportunity attack and walk right past you in search of a squishier target.
The most important consideration when looking at your proposed changes, as well as the DM-specific advice, is "are we having fun?" The DM shouldn't be out to get your character; if they wanted to "win", they could just put you up against a swarm of low-level druids with entangle. That said, most players (you included, it seems) want an interesting challenge rather than a steamroll.
I recommend that your DM tries out some of these less-extreme changes for a couple of sessions. If more fun is had, then great! If not, then maybe it's time to look at a different build - an Eldritch Knight fighter or a Hexblade warlock are less extreme possibilities for a melee caster.