there are differences in formae speciales between F. oxysporum. Please read the following articles for further information. F. oxysporum forma specialis (f.sp.) lycopersici attacks tomato, the F. oxysporum f.sp. cubense attacks the banana etc... However the host range does not necessarily meet with the name. Some f.sp. could have a broader host range. However F. oxysporum f.sp. capsici is the main cause of wilting on pepper.
Please have a look at the following material for further information:
Article Characterization and Pathogenicity of Fusarium oxysporum as ...
Article Antifungal exploitation of fungicides against Fusarium oxysp...
Article Occurrence of Fusarium oxysporum causing wilt on pepper in Mexico
Article The basis of host recognition in Fusarium oxysporum f. sp. Lycopersici
Same Fusarium spp. causes wilt in hot pepper , tomato and other plants need to prove with the help of phylogenetic relationship. I am not sure that same Fusarium spp causes wilt in different plants. Molecular identification and phylogeny may give solution.