Subaru's self harm being canonized makes this whole scene sad in retrospect. Yes, Beako was clinging to him because she was dealing with severe separation anxiety but she also did it because she’s genuinely terrified of what he’ll do to himself if she doesn't stay glued to him ☹️
I bet most anime only fans missed this detail but Emilia knew Subaru was acting weird because he called her "Emilia-chan" instead of "-tan." Since he was hiding his amnesia, dropping his usual nickname felt like a bizarre, unfunny joke to her, which is why she got so worried.
Fun fact for anime only but Subaru is indeed part of #shtwt. The anime skipped over it before but Subaru’s self harm habits first came to light in Arc 2, after being killed by the twins, which only Beako knew of and he still has that habit to date whenever he's anxious