Do all plant hormones of one group show the same qualitative effect with respect to their target pathway(s) and the biological response?
Preface: As @David says, the question is very broad. So, here, I'll take up the most common plant growth factors i.e. auxins to give you an idea.

Short Answer: We don't definitely know yet whether the pathways of two auxins (I take indole-3-acetic acid (IAA) and indole-3-butyric acid (IBA) here) are exactly same or not.

Background: We know that auxins have very similar functions1:


  Auxins are a class of plant hormones (or plant growth substances) with some morphogen-like characteristics. Auxins have a cardinal role in coordination of many growth and behavioral processes in the plant's life cycle and are essential for plant body development.


So much similar that2:


  genetic evidence has been found that suggests that IBA may be converted into IAA through a similar process to $\beta$-oxidation of fatty acids.


Now, when it comes to whether the exact mechanisms of how IAA and IBA work are similar or not then, as I said, we don't know yet.

When we talk about IAA, there is a theory known as Acid Growth Theory3 (though there are evidences against it):


  Plant cells elongate irreversibly only when load-bearing bonds in the walls are cleaved. Auxin causes the elongation of stem and coleoptile cells by promoting wall loosening via cleavage of these bonds. This process may be coupled with the intercalation of new cell wall polymers. Because the primary site of auxin action appears to be the plasma membrane or some intracellular site, and wall loosening is extracellular, there must be communication between the protoplast and the wall. Some "wall-loosening factor" must be exported from auxin-impacted cells, which sets into motion the wall loosening events. About 20 years ago, it was suggested that the wall-loosening factor is hydrogen ions. This idea and subsequent supporting data gave rise to the Acid Growth Theory, which states that when exposed to auxin, susceptible cells excrete protons into the wall (apoplast) at an enhanced rate, resulting in a decrease in apoplastic pH. The lowered wall pH then activates wall-loosening processes, the precise nature of which is unknown. Because exogenous acid causes a transient (1-4 h) increase in growth rate, auxin must also mediate events in addition to wall acidification for growth to continue for an extended period of time. These events may include osmoregulation, cell wall synthesis, and maintenance of the capacity of walls to undergo acid-induced wall loosening. At present, we do not know if these phenomena are tightly coupled to wall acidification or if they are the products of multiple independent signal transduction pathways.


However, there are evidences supporting this hypothesis, like this one4:


  When auxin stimulates rapid cell elongation growth of cereal coleoptiles, it causes a degradation of 1,3:1,4-$\beta$-glucan in hemicellulosic polysaccharides. We examined gene expressions of endo-1,3:1,4-$\beta$-glucanase (EI) and exo-$\beta$-glucanase (ExoII), of which optimum pH are about 5, and molecular distribution of hemicellulosic polysaccharides in barley (Hordeum vulgare L.) coleoptile segments treated with or without IAA. IAA (10–5 M) stimulated the gene expression of EI, while it did not affect that of ExoII. IAA induced gene expression of EI after 4 h and increased wall-bound glucanase activity after 8 h. The molecular weight distribution of hemicellulosic polysaccharides from coleoptile cell walls was shifted to lower molecular weight region by 2 h of IAA treatment. Fusicoccin (10–6 M) mimicked IAA-induced elongation growth and the decrease in molecular weight of hemicellulosic 1,3:1,4-$\beta$-glucan of coleoptiles in the first 4 h, but it did not promote elongation growth thereafter. These facts suggest that acidification of barley cell walls by IAA action enhances pre-existing cell wall-bound glucanase activity in the early first phase of IAA-induced growth and the late second phase involves the gene expression of EI by IAA.


But when it comes to IBA, all we have is this2:


  Although the exact method of how IBA works is still largely unknown, genetic evidence has been found that suggests that IBA may be converted into IAA through a similar process to $\beta$-oxidation of fatty acids. The conversion of IBA to IAA then suggests that IBA works as a storage sink for IAA in plants. There is other evidence that suggests that IBA is not converted to IAA but acts as an auxin on its own.


So, if IBA is converted to IAA before use then, obviously, it works the same way. But the article below suggests that either the mechanism or effectiveness of IAA and IBA is different5:


  We have examined in vitro rooting of apple ‘Jork 9‘ shoots exposed for three weeks to each of the three auxins commonly used for ex vitro rooting: indole-3-acetic acid (IAA), indole-3-butyric acid (IBA) and $\alpha$-naphthaleneacetic acid (NAA). During the initial five days of the rooting treatment, the cultures were incubated in darkness. In this period, the root initials are formed. Then, the cultures were moved to the light. NAA resulted in a low (ca. 8 roots), and IAA or IBA in a high (ca. 15 roots) maximal root number. The maximal root number was reached at a wide range of IAA concentrations (10-100 $\mu$M) but at only one concentration of IBA (10 $\mu$M) or NAA (3 $\mu$M). With NAA and IBA, growth of roots and shoots was much more inhibited than with IAA. For these reasons, IAA is the preferable auxin for in vitro rooting of apple ‘Jork 9’ shoots.


This suggests that IAA and IBA use different mechanisms or that IBA can't use the same mechanism as effectively as IAA.

References:


Auxin - Wikipedia
Indole-3-Butyric Acid - Wikipedia
The Acid Growth Theory of auxin-induced cell elongation is alive and well
Auxin-Induced Elongation Growth and Expressions of Cell Wall-Bound Exo- and Endo-$\beta$-Glucanases in Barley Coleoptiles
Effectiveness of indoleacetic acid, indolebutyric acid and naphthaleneacetic acid during adventitious root formation in vitro in Malus ‘Jork 9’