What exact function do に を serve here together?
"His gaze full of trust toward me reminds me of my childhood"?
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That を is a normal usage as the object marker. 子供時代 is the direct object of the verb 思い出す. The speaker remembers his childhood.
The に after 眼差し is somewhat literary. It is used for something that triggers a certain emotion in the agent (or doer) of the verb. It’s related to this. In this particular case, the gaze reminds the speaker of his childhood.
The に after 俺 seems wrong to me. It should be を.