Although a growing body of research has examined students' attitudes toward generative artificial intelligence (GenAI) in higher education, few studies have compared perceptions across contrasting institutional contexts or explored how students' reported uses of GenAI relate to broader learning practices. This study addresses that gap by examining university students' perceptions, self-reported competence, and use of GenAI at two Swedish universities with different academic profiles: a technology-oriented institution and a broader multidisciplinary institution. The study is based on an exploratory questionnaire survey administered to all enrolled students at both universities, yielding 1,097 responses (University A response rate: 11.27%, University B response rate: 14.06%) from students across diverse disciplines, including engineering, nursing, and criminology. Quantitative data were analyzed using reliability analysis, exploratory factor analysis, and non-parametric group comparisons, supplemented by thematic analysis of qualitative responses. The analysis identified three reliable constructs: perceived learning benefit, perceived institutional support and integration, and self-reported technical knowledge and competence. Across both institutions, students reported generally positive attitudes toward GenAI and described using it primarily for information retrieval, text refinement, and text analysis, but also as a discussion partner or personal tutor in ways that suggest both surface-level and more dialogic forms of engagement. Comparisons between the two universities showed broad similarity across most measures, with the only statistically significant difference relating to perceived institutional support and integration, which was rated higher by students at the technology-oriented university. Students at both institutions also viewed GenAI primarily as a complement to, rather than a replacement for, traditional teaching, while reporting only moderate trust in AI-generated outputs. These findings suggest that GenAI is already embedded in students' study practices, but that its use is largely self-directed rather than strongly shaped by institutional context. The study thus contributes comparative empirical evidence on student engagement with GenAI across contrasting higher education settings and highlights the need for pedagogical and institutional strategies that support critical, reflective, and responsible use.