If the dependent variable is a Poisson distribution (related to a counting process like the number of accidents, the number of reports, the number of patients, ...), you can use the Poisson regression model. As you know, the DV in regression needs to be normal. To meet this presumption, we transform DV with the logarithm function. It is well known that the IVs are normal, and this presumption is necessary too. In addition, residuals must be uncorrelated. Check out these requisites using plots and tests.
It is well known that survival analysis is a branch of statistics to analyze the expected duration of time until one event occurs, such as death in medicine. I guess you mean Kaplan-Meier Survival Analysis or Kaplan-Meier Estimate, which based on lifetime data (time to event data) is looking for estimation of the survival function. This curve represents the pattern of decreasing the sample under study expecting the event. Notably, this estimate is non-parametric and based on no presumption.
Cox regression model (Cox Proportional-Hazards Model) is a model to link the survival function (hazard function) of two groups. It is based on a presumption about the ratio of the hazard functions.
With Poisson regression, the response variable of interest is a count (or possibly a rate). With Cox regression (or alternative modelling strategies from survival analysis), the response variable is the time that has elapsed between some origin and an event of interest. https://stats.stackexchange.com/questions/44046/what-are-the-differences-between-survival-analysis-and-poisson-regression
Kaplan–Meier provides a method for estimating the survival curve, the log rank test provides a statistical comparison of two groups, and Cox's proportional hazards model allows additional covariates to be included. Both of the latter two methods assume that the hazard ratio comparing two groups is constant over time. https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1065034/