Since in Latin no compound form of verb tense exists, AFAIK, I thought that origin of Present Perfect should be sought in Proto-Germanic also for Romance languages, but I found out that Present Perfect exists in Romanian as well and I think that Germanic influences should not have affected Romanian language. Should instead the origin be sought in Proto-Germanic nonetheless?
The similarity is due to a common pathway of grammaticalistion. The have + past participle form comes from a resultative construction (Bybee, Perkins and Pagliuca, 1994), which commonly leads to the perfect. Note that Bybee et al. do not use the term resultative in the usual complex-predicate sense, but use it to mean sentences like 'The door is opened', whereby a past action leads to a present state.
The English have + past participle construction comes from sentences like (1), where have was still possessive and the participle agreed with the object in number, case and gender:
(1) Ic haefde hine gebundenne. I had him bound I had him in a state of being bound. (Traugott 1972, cited in Bybee et al.)
This construction later became the perfect, which asserts that a past action has effect on the present - a similar meaning to the resultative. The participle now no longer agrees with the object and the auxiliary and partiicple should usually be adjacent.
In the Romance case, the resultative was already present in Vulgar Latin, both in have + participle form and in participle + have form:
(2) Metuo enim ne ibi vos habebam fatigatos. Fear.1sg for lest there you have-IMPF-I SG tired 'For I fear that I have tired you.' (early fifth century, Augustine; Fleischman 1982, cited in Hopper and Traugott ) (3) in ea provincia pecunias magnas collocatas habent in that province capital great invested have.3pl They have great capital invested in that province.
Again, the construction became the perfect (and, in modern French, the past perfective). It lost the object agreement, spread to all dynamic verbs (not just change-of-state), and the have + participle form was the only form that survived in languages like French.
There is sufficient semantic similarity between the resultative and the perfect - described by Comrie thus: 'the present auxiliary conveys the present meaning, while the past participle conveys that of past action' - and this would explain why both Germanic and Romance languages developed this form.
Bybee, J. L., Perkins, R. D., & Pagliuca, W. (1994). The evolution of grammar: Tense, aspect, and modality in the languages of the world (Vol. 196). Chicago: University of Chicago Press.
Comrie, B. (1976). Aspect: An introduction to the study of verbal aspect and related problems (Vol. 2). Cambridge university press.
Hopper, P. J., & Traugott, E. C. (2003). Grammaticalization. Cambridge University Press.
To the excellent answer by @WavesWashSands I'll only add that some Latin verbs employed a perfective construction with the verb esse "to be" and a participle, which at some point could have motivated the appearance of the other well-known pattern for the compound perfect as found in Italian and French, for example:
- Siamo arrivati. "We have arrived." (lit. "We are arrived.")
- Je suis tombé. "I fell, I have fallen." (lit. "I am fallen.")
Note that the participle agrees with the subject in number and gender (e. g. if the first sentence were uttered by a woman, it would be Sono arrivata, with -a marking female-singular).
Since most verbs did not use this pattern, in other Romance languages (such as Spanish) the form derived from habēre gradually displaced the form derived from esse and all the compound perfect tenses came to be conjugated using the latter (regularization).
Also, as the Latin verb tenēre "to keep" gradually took the place of habēre for the meaning "to have", some dialects also adopted it as an auxiliary. To this day Portuguese employs ter ("to have", cognate with Spanish tener) as the preferred auxiliary for compound verbs, though the older form haver is still understood.
Finally, be aware that Latin habēre is not cognate with English have (or better, with its Germanic ancestor) despite the phonetical similarity. Proto-Germanic *habjaną is cognate with Latin capere "to take, to seize", whence Spanish caber "to fit".