No.
There are some verbs that can fit both roles but you look at them differently.
https://www.usingenglish.com/glossary/stative-verb.html:
Stative verbs are
verbs that describe a state and consequently the stative verb is not usually used in the
progressive aspect, which is used for incomplete actions in progress. STATIVE VERB EXAMPLE:
EG: They
own a cottage in Somerset. (The possession is a state and not an action. We cannot write this sentence in the progressive aspect)
They are in contrast to action verbs, where something happens. (buy, eat, swim, yell, etc.)
https://www.usingenglish.com/glossary/linking-verb.html: A
linking verb is a
verb that connects the
subject to the
complement. They are sometimes called copula or copular verbs. EG: That food smells good.
('Smells' connects the subject to the
adjective that describes it.)
The following are the principal
linking verbs in English that can be used to connect the
subject to an
adjective:
Be; Look; Feel; Taste; Smell; Sound; Seem; Appear; Get; Become; Grow; Stay; Keep; Turn; Prove; Go; Remain; Resemble; Run; Lie