The verb Russian drops that Serbian refuses to
„Ја сам студент.“ carries a verb Russian's equivalent sentence does not: Я студент has none at all. Serbian's present-tense to be has never gone silent; Russian's did, centuries ago. The gap explains why dropping сам is the single most persistent mistake a Russian speaker makes in Serbian - and why the instinct behind it is not a mistake at all in the language it came from.
Two sentences say the same thing, and one of them is missing a word. „Ја сам студент.“ needs its verb about as obviously as an English sentence would - and then „Я студент.“ turns up with no verb at all, and no Russian speaker feels the gap for a second. This blog has already told the story of two things Serbian kept that Russian let go quiet: the aorist tense and the vocative case. The present-tense copula belongs in the same small collection, and it is the one a beginner runs into on day one.
Common Slavic had the verb in both languages
Old Church Slavonic conjugated to be in the present tense for every person, in every Slavic language's ancestor alike - the verb was never optional. Serbian's short present-tense forms are that very same conjugation, worn down by a thousand years of sound change but still doing the same job they always did (the full paradigm is in the бити lesson linked below). Old Russian had an equivalent set of forms too; the difference is not that Serbian innovated a verb Russian never had. It is that Russian dropped one.
Where Russian's copula actually went
Russian did not lose the verb быть altogether - it conjugates normally in the past and future, and one fossilized present-tense form, есть, survives in a narrow existential use (as in „Есть время.“, there is time). What disappeared is specifically the present-tense copula linking a subject to a noun or an adjective, and it disappeared gradually over several centuries, roughly finishing the retreat by the eighteenth century. Nothing comparable happened in Serbian: the present tense of бити stayed exactly where it started, doing exactly the job it always did.
On je student, a ja sam lekar.
He's a student, and I'm a doctor.
Two clauses, two copulas - Serbian repeats је/сам for each subject; a word-for-word Russian translation has neither.
Jesi li ti lekar?
Are you a doctor?
The question needs the long form јеси; Russian needs no verb at all, only the rising question intonation.
The trap hiding inside есть
The surviving Russian есть is not a copula in exile, ready to reappear once a Serbian sentence needs a verb - it means there is/are, not is, and it does not conjugate for person. A Russian speaker reaching for it as a stand-in for the missing сам produces a sentence that means something else entirely, not simply an old-fashioned or formal one. The two functions look related because they share a common ancestor, but in Serbian they are two entirely different words doing two entirely different jobs: бити for identity and description, имати (with its own existential use, нема никог - there's nobody) for possession and existence.
Нема (there isn't/aren't) is the negative of имати, not of бити - it answers does X exist, never is Y a doctor. Keep the two verbs' negative forms apart: нисам negates identity and description, немам negates possession, and нема negates existence. Reaching for the wrong one produces a grammatical sentence that answers a question nobody asked.
Not unique to Serbian, either
Zero-copula present tense is actually the unusual pattern across the Slavic family, not the common one: Polish, Czech, Bulgarian and Serbian itself all keep some overt present-tense to be as the obligatory rule. Seen from that angle, Serbian is not the exception needing an explanation - Russian is, and a Russian speaker's instinct to drop the verb is simply the instinct of an outlier within its own family, not a shortcut every language shares.