feat: store server_info on ClientSession after initialization #298

Job	Run time
checks / readme-snippets	11s
checks / pre-commit	38s
checks / test (3.12, locked, windows-latest)	3m 29s
checks / test (3.11, locked, ubuntu-latest)	44s
checks / test (3.11, lowest-direct, ubuntu-latest)	43s
checks / test (3.12, lowest-direct, ubuntu-latest)	50s
checks / test (3.11, locked, windows-latest)	3m 35s
checks / test (3.10, locked, ubuntu-latest)	46s
checks / test (3.12, locked, ubuntu-latest)	50s
checks / test (3.10, lowest-direct, ubuntu-latest)	47s
checks / test (3.11, lowest-direct, windows-latest)	2m 37s
checks / test (3.12, lowest-direct, windows-latest)	3m 42s
checks / test (3.13, lowest-direct, windows-latest)	2m 42s
checks / test (3.13, locked, ubuntu-latest)	45s
checks / test (3.10, locked, windows-latest)	2m 56s
checks / test (3.10, lowest-direct, windows-latest)	2m 36s
checks / test (3.14, locked, ubuntu-latest)	2m 21s
checks / test (3.14, lowest-direct, windows-latest)	2m 29s
checks / test (3.14, locked, windows-latest)	3m 35s
checks / test (3.14, lowest-direct, ubuntu-latest)	1m 14s
checks / test (3.13, locked, windows-latest)	2m 24s
checks / test (3.13, lowest-direct, ubuntu-latest)	43s
all-green	4s
	40m 41s

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