If a company were to pursue different project alternatives in which to invest, would I perform a SWOT analysis on each project alternative or should I do a SWOT analysis of the company and, based on that, make a decision about the different alternatives I have?
SWOT Establishes a Business Case, Not Comparative Value
In general, (S)trengths and (W)eaknesses in a SWOT diagram are about the company itself, while (O)pportunities and (T)hreats are usually about the market. A project often presents an opportunity or mitigates a threat. Likewise, companies have strengths and weaknesses that can make a given project more or less of a natural fit for the organization.
I've certainly seen people use SWOT diagrams to promote a project or initiative, but in the general case it isn't really an ideal tool for portfolio management. It may help to establish a business case, but doesn't intrinsically create a metric for comparing projects against one another or quantifying value.
Portfolio management is a discipline, and as such there is no One True Way™ to manage a portfolio. However, a solid mix of qualitative and quantitative criteria, stakeholder communication and buy-in, and weighted analyses using company-specific metrics are all essential elements of successful portfolio management.
An investment alternative or project alternative itself does not have strengths and weaknesses per se. What it produces as an outcome would have strengths and weaknesses. From an evaluation standpoint for selection, you would perform a benefit / cost / risk analysis, which is sort of way to predict its outcome SWOT, I suppose. I think in most cases, when selecting from alternatives, folks reference the strategic analysis as a cost / benefit exercise or a ROI v. a SWOT.