First of all, if you want to consider specifically one "part", "branch", or "tradition" of philosophy and not philosophy as such, you need some sort of characterisation in order to specify this part of philosophy and to set it apart from other parts of philosophy. This usually is accomplished by using adjectives like "Latin philosophy", "Christian philosophy", "German philosophy", "Greek philosophy", "ancient philosophy", and so on. The question is only which adjective or characterisation you should choose and which adjective reflects most aptly what you want to determine. This is often simply a matter of taste.
With regard to the tradition I think you mean by "Islamic philosophy", there are usually three options for naming that tradition:
- Arab philosophy
- Arabic philosophy
- Islamic philosophy
The first, Arab philosophy, is rarely used. In addition, it is also misleading. Not many philosophers of the tradition were Arabs, given that "Arab" is taken to signify a geographical origin, i.e., people from the Arabian Peninsula. Many philosophers of the tradition were from Persia or Andalusia, which clearly are not on the Arabian peninsula. If "Arab" is taken to signify a language, then "Arabic" is the more proper adjective, which brings us the the next option.
The second, Arabic philosophy, is used if one wants to draw attention to the language in which the works of that tradition are primarily written. This would be the Arabic language. Some works, however, are written in Persian or Syriac. This is why some say that "Arabic philosophy" signifies not so much the language in which the works are written but more precisely refers to that tradition of philosophy which had its origin in the translation of Greek scientific and philosophical works (often through Syriac) into the Arabic language. This translation movement began roughly in the 7th century. So, "Arabic philosophy" would be a designation that includes even Syriac and Persian works, because all these works are the result of a tradition that began with (or at least was very fundamentally influenced by) a translation of Greek philosophical works into Arabic. It would also include the works of Jews and Christians, because they all participated in that same endeavour that began with the translations from Greek into Arabic. The translations, by the way, were often produced by bilingual Christians first from Greek into Syriac and then from Syriac into Arabic.
The third, Islamic philosophy, describes the cultural context in which this branch of philosophy originated and was cultivated. "Islamic" is, thus, not primarily meant to be a religious term but a cultural one. Since some philosophers in that tradition, however, were Christians or Jews, people tend to modify the designation and say "philosophy in the Islamic world" or something like that. This emphasises the cultural aspect even more and so includes the Eastern parts of Persia up to India as well as Syria, the Arabian peninsula, North Africa and Southern Spain. Moreover, again, works written in Persian and Syriac or written by Christians and Jews would also be part of that tradition.
These are roughly the three options one has if one wanted to find a name for that philosophical tradition. The second and the third are both valid suggestions.
Something else that is important: Greek texts and their Arabic translations were a major source of inspiration for Islamic or Arabic philosophy, but they were not the only one. This is why some people don't like "Arabic philosophy" as a name for that tradition, because this, they say, emphasises too much the language aspect and, thus, the Greek influence. These people would rather choose "Islamic philosophy". Indeed, it is clear that Arabic or Islamic philosophy is more than a simple continuation of Greek thought.
So, again, it is a matter of taste. But as long as the above mentioned aspects are understood, it is not so important anymore whether you pick option two or three.
Finally, let me say that I don't share your view that "philosophers also don't have any concern to religion". This is not true. One fascinating aspect about Arabic/Islamic philosophy is precisely the unique interplay between Greek pagan or Greek Christian thought that is discussed and reinvented by Muslim intellectuals often (in varying degrees) with an eye on religious and specifically Islamic concerns that you can find in the texts within this tradition.