The verb 'bear' descends from Old English 'beran' and PIE *bʰer- (to carry), one of the most productive roots in Indo-European — the same root behind Latin 'ferre,' Greek 'phérein,' and English derivatives from 'birth' to 'transfer' to 'suffer,' all united by the core idea of carrying.
To carry, support, or endure; to bring forth or give birth to; to hold up under weight or pressure.
From Old English 'beran' (to carry, bring, bear), from Proto-Germanic *beraną, from PIE *bʰer- meaning 'to carry, to bear.' This is one of the most prolific roots in Indo-European, producing Latin 'ferre' (to carry), Greek 'phérein' (to carry), Sanskrit 'bhárati' (carries), and an enormous family of derivatives including 'transfer,' 'fertile,' 'suffer,' 'differ,' and 'conference.' The semantic range — carry, endure, give birth — was already present in PIE. Key
The verb 'bear' and the noun 'bear' (the animal) are completely unrelated etymologically. The verb comes from PIE *bʰer- (to carry), while the animal name comes from Proto-Germanic *berô (the brown one) — a taboo replacement for the original PIE bear-word *h₂ŕ̥tḱos, which speakers avoided saying aloud for fear of summoning the animal.