If I take sphere and remove two open disks from it and on the boundary of that space I make identification like on the picture, what do I get? Are both of those objects Klein's bottles?
This is what I want: 
]1
2026-04-03 03:41:16.1775187676
Connected sum of two "same" Klein bottles
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A $2$-sphere with two disks removed is homeomorphic to a cylinder $I\times S^1$.
If I understand your picture correct you now want to identify the two boundary components of this cylinder. In the way it is pictured you get a Klein bottle.