In its discussion of Boneh (lit. building) among the thirty nine prohibited actions on Shabbat, the Gemara (Shabbat 47a) analyzes the permissibility of assembling a cot on Shabbat. Based on this discussion the Shulhan Aruch (O.H. 313:6) rules that if the pieces are interlocked and affixed strongly, which the Gemara refers to as “Taka” (lit. affixed), then the assembly would be considered Boneh and would be forbidden. If, however, the parts are assembled in a loose manner it would be permitted. The Mishna Berura (ibid:45) quotes the Magen Avraham and the Taz, who both discuss assembled cups and write that since they are constantly assembled and disassembled as part of their normal use, one would be able to put such a cup together on Shabbat. Similarly, the normal way to use a drink bottle, for example, is to remove the cap and to put it back on regularly, and thus would be permitted. Regarding objects which are not regularly opened and closed, such as a salt shaker, Rabbi Shlomo Zalman Auerubach ( Minchat Shlomo pg 70) writes that since it has the ability to be readily opened and closed, doing so on Shabbat would be permitted as well.
With regards to tightening a screw on Shabbat, one should be careful as this action can be a violation of Boneh on a biblical level. (c.f Tosfot Shabbat 47a). As such, Rabbi Baruch Toledano (Kitzur Shulhan Aruch, 291:6 ) adds that when one is assembling something with screws, one must screw them tightly so as to minimize the possibility of needing to tighten them on Shabbat. Another potential issue is assembling a needle or syringe on Shabbat. Certainly, if the syringe is needed for something that is Pikuah Nefesh (lit. life-preserving), such as insulin for a diabetic, then it is permitted. Otherwise, putting together the syringe is considered Boneh and potentially also the forbidden act of Makeh BePatish (the finishing touches when creating an object).
Therefore, although Boneh classically applies to structures built on the ground, it can also apply to objects that are built when different parts are assembled in a sturdy manner. Some objects are assembled with little force but because of the mechanism of the parts, the attachment is very strong, such as belt buckles which are easily clipped onto a belt but remain sturdily attached. Regarding these kinds of objects, I heard from Rabbi Yosef Shalom Elyashiv that since the parts are not put together forcefully, they do not have the status of Taka, and one would be able to put it together on Shabbat.
Summary: One should be careful on Shabbat with objects that require assembly as one may violate the act of Boneh.