EDIT: Simpler and cleaner approach could be to use ggh4x::facet_nested.

...
scale_y_continuous(breaks = NULL) +
coord_cartesian(clip = "off") +
ggh4x::facet_nested(S1 + Similar ~ ., scales = "free_y")

(For a subset of the data)

I’m curious if there’s a simpler hack for this. I used tidytext::reorder_within, which creates facet-specific axis definitions/orderings, and tidytext::scale_y_reordered to display those as unadorned labels.

db0 |> 
  ggplot(aes(y = tidytext::reorder_within(
               paste(Similar, y_position, sep = "___"), y_position, S1),
             group = interaction(Similar, Model))) + 
  ...
  tidytext::scale_y_reordered() +
  coord_cartesian(clip = "off") + # to keep labels outside facets
  geom_text(aes(x = (Altered + Unaltered) / 2,
                label = Model), size = 4, vjust = -0.2,
            check_overlap = TRUE) +
  facet_grid(S1 ~ ., scales = "free_y")

Personally I think that the approach by @JonSping using ggh4x::facet_nested makes your plot much easier to read.

Anyway here is an approach to have the labels on the axis which simply sticks with a discrete scale, i.e. maps Similar on y and computes the y_positions for the geoms per panel using a split-and-bind approach where to simplify the computation I use scales::rescale instead of a bunch of ifelse.

library(ggplot2)
library(dplyr, warn = FALSE)

set.seed(123)

# Dodge width
dw <- .9

db1 <- db0 |>
  split(~S1) |>
  lapply((x) {
    n_models <- n_distinct(x$Model)
    dw <- dw / n_models
    x |>
      mutate(
        y_position = as.numeric(factor(Similar)) + dw * scales::rescale(
          as.numeric(factor(Model)),
          to = c(-1, 1)
        ),
        Increase = ifelse(Altered > Unaltered, TRUE, FALSE)
      )
  }) |>
  bind_rows()

db1 |>
  ggplot(aes(y = Similar, group = interaction(Similar, Model))) +
  # Keep the discrete y scale
  geom_blank() +
  geom_segment(aes(
    y = y_position, yend = y_position,
    x = Unaltered, xend = Altered, color = Increase
  )) +
  geom_point(aes(y = y_position, x = Unaltered), color = "black") +
  geom_point(aes(y = y_position, x = Altered, color = Increase)) +
  scale_color_manual(values = c("purple", "gold")) +
  scale_x_continuous(name = "Value", limits = c(0, 1)) +
  theme_test() +
  theme(
    axis.text.y = element_text(size = 10),
    panel.grid.major.y = element_line(color = "grey90"),
    panel.grid.minor.y = element_blank(),
    axis.title.x = element_blank(),
    axis.title.y = element_blank(),
    legend.position = "none"
  ) +
  geom_text(
    aes(
      x = (Altered + Unaltered) / 2,
      y = y_position + .05,
      label = Model
    ),
    check_overlap = TRUE,
    vjust = 0,
    size = 8 / .pt
  ) +
  facet_grid(S1 ~ ., scales = "free_y") +
  coord_cartesian(clip = "off")