Fingir and pretender are not synonyms, but pretender could translate as to pretend but only in very limited circumstances. Pretend has several meanings in English, and some of them could indeed be used for the meaning of pretender which the DLE defines thus:
1. tr. Querer ser o conseguir algo.
2. tr. Hacer diligencias para conseguir algo.
3. tr. Dicho de una persona: Cortejar a otra.
So it can be want to be or acquire something, to do what is necessary to get something, or to court someone. Let's compare with the OED's definitions for to pretend:
a. intransitive. figurative. To tend to an end or point in action, speech, etc. Also: to extend in time. Obsolete.
b. intransitive. To stretch or reach forward; to move or go forward; to direct one's course; to tend. Usually with to or for. Obsolete.
†10. transitive. To intend, plan.
a. With infinitive as object. Obsolete.
b. With clause as object. Obsolete. rare.
c. With simple object. Obsolete.
†a. intransitive. To aspire; to have pretensions. Chiefly with to. Obsolete.
b. transitive. With infinitive as object. To aspire, presume; to venture; to try, attempt. Now rare.
†c. intransitive. To pay court to (a prospective spouse); to seek to be married to (a person). Obsolete.
Example sentences from the OED show sentences that would be translated with pretender:
- For to what fyn he wolde anon pretende, Þat knowe ich wel
Aquel fin que él pretendería pronto, el cual sé muy bien,
- I wyll pretende To stey to my father.
Pretenderé quedarme con mi padre.
- I find by experience that we cannot acquire that end which is pretended to by such addresses.
Noto por experience que no podemos conseguir aquel fin que se pretenden tales discursos.
- We pretend, that this City, already famous for the Defeat of two of your Armadas, shall become far more so by the Disgrace of this your third.
Pretendemos que esta ciudad, ya famosa por la derrota de dos de tus armadas, sea aún más famosa por la desgracia de esta tercera tuya.
- All oþir askiþ & sekiþ her ovne comodites; þou pretendist allone myn helþe & my profityng
Los demás piden y buscan sus propias comodidades; tú pretendes solo mi ayuda y mi beneficio
- I am not such a fool as to pretend to you now I am poor, and you have got altogether above me.
No sé tan tanto para pretenderte ahora que soy pobre, y tienes todo por encima de mí.
- It is difficult to forecast the feeling of Congress, and few pretend to guess the probable terms.
Es difícil predecir los sentimientos del Congreso, y pocos pretenden adivinar los términos probables.
- Some pretende to hye estates & grete richesses, & other ben content with lytil estate.
Algunos pretenden altas condiciones y grandes riquezas, y otros están contentos con bajas condiciones.
- That..Step..lays her under the Foot of the Man she pretends to.
Aquel...paso... le pone debajo del pie del hombre que ella pretende.
However, as you'll no doubt notice, both between the definitions and the sheer spelling of the English, the use of pretend in this manner is rare today, almost completely usurped by its usage equivalent to fingir. So while Google is technically correct in equating pretend and pretender, it's not very helpful, and it's flat out wrong to say that finger and pretender are synonymous.