Is the word clínica the same in Brazilian and European Portuguese?

Yes

The spelling, meaning, and grammatical function of "clínica" are identical in both Brazilian and Continental Portuguese. The only difference is pronunciation. In Brazilian Portuguese, the vowels are generally more open and clearly articulated. In Continental Portuguese, unstressed vowels tend to be "reduced" or "clipped," meaning the pronunciation sounds more closed and the rhythm is more stress-timed.

Brazilian Portuguese Examples

1. Preciso marcar uma consulta na clínica. (I need to schedule an appointment at the clinic.)
2. A clínica fica perto do metrô. (The clinic is near the subway.)
3. Ela trabalha numa clínica de estética. (She works in an aesthetics clinic.)
4. A clínica odontológica é muito boa. (The dental clinic is very good.)
5. A clínica está com uma promoção ótima. (The clinic has a great sale/promotion.)

Continental Portuguese Examples

1. Preciso marcar uma consulta na clínica. (I need to schedule an appointment at the clinic.)
2. A clínica fica perto do metro. (The clinic is near the metro.)
3. Ela trabalha numa clínica de estética. (She works in an aesthetics clinic.)
4. A clínica dentária é muito boa. (The dental clinic is very good.)
5. A clínica está com uma campanha de descontos. (The clinic has a discount campaign.)