This is not balanced, this is broken
This has another problem than mere power balance: it is a one trick pony that will not play well. In my play experience as well as in that of others, managing too many minions is a chore and detracts from fun. This class can essentially do very little other then creating an ever-bigger stable of minons that you then need to manage.
Granted, the spell list is trying to address this and create some other options by adding a few "best-of" spells that are not just "summon another kind of creature" like healing spirit, misty step, dimension door, forcecage and teleport, but considering the overwhelming focus on just summoning stuff, these feel both a bit out of place and cheesy.
Unbalanced Exploits
For your main question, this class is not only unbalanced in regards to how it plays but also regarding power level. The main class is probably fine, but in combination with the Collector subclass, it is not. The main issues I see:
Seize control is broken. This allows you to use your Summon Monster spell to summon any kind of creature, provided you fought and killed it before.
Even ignoring special abilities, a CR 2 monster is a deadly encounter for a third level party -- a though fight that could well kill a character. Since Seize Control gets rid of the CR level ceiling based on your caster level, you could with some luck defeat one of them, and then have a monster fight along you that is nearly as dangerous as your entire team. Note that you could get these at level 3, because the fit into your Proficiency Bonus 2 ceiling.
But that is not even the main issue here. There is a reason polymorph does not allow you to create targets other than beasts: monsters have all kinds of insane special abilities that would be utterly unbalancing in the hand of player characters. Normally, the new Summon Monster spell introduced here has a safety valve in that the DM picks what you get, but with this feature, you can pick and choose yourself, and you over time can get anything up to CR6. Here is a nice selection of the things you could get.
- How about a pixie, CR 1/4, that provides polymorph, while you are level 3?
- How about getting a rust monster CR 1/2 and destroying metal vault doors with ease?
- How about an Intellect Devourer, CR2, that can stun any creature that fails a DC 12 Intelligence save? Want to take out a CR 17 Goristo demon? No problem, they only have Int 6. you can even have the Devourer Body thief them and control their body.
- How about a wererat, also CR2, that is immune to non-magical, unsilvered weapons and that most tier one opponets will not be able to hurt at all?
- How about a unicorn, CR4, that can teleport you and your pals, but does not need any spellcasting feature for that. Teleport while at level 7 is surely nice, no?
And so on, you get the idea. Then, once you get to level 6, and get access to Improved Shunt, you can build a whole zoo of such monsters, like a bag of golf clubs, have them all in storage and pick the right tool for whatever challenge you might face. How is your DM going to avoid that? Never let you encounter any monsters with special abilities?
I'm not even getting into the Mystic Callings that you can have a likewise unlimited number of after level 6, one on each creature stored, like the Passageway of the Astral Wayfarer which allows you to summon the a creature you shunted for free while casting a full other spell or making full other use of your action. Or Throng of Allies for nova actions during boss fights -- just build up a dozen monsters in shunt. Who cares if they are gone at the end of their first round, by then the opponent is long dead. There are many more issues.
This class is a quagmire of abuse and problems.