No way to sign in multiple github accounts

### Description

I used to be able to sign in to multiple github accounts and have the UI pick the right account. Now it seems I can only sign in with a single account, and this seems to get out of sync with what "gh" thinks I'm signed in with.

### Issue Details

Item|Value
---|---
Issue Category|Bug
Target|Visual Studio Code
VS Code Version|Code 1.132.0 (df53daabb18cd157bdb08c7f01c34df936cf12f4, 2026-08-04T15:30:20Z)
OS Version|Windows_NT x64 10.0.26220

System Info

Item|Value
---|---
VS Code Version|Code 1.132.0 (df53daabb18cd157bdb08c7f01c34df936cf12f4, 2026-08-04T15:30:20Z)
OS Version|Windows_NT x64 10.0.26220
CPUs|Intel(R) Core(TM) i7-9700K CPU @ 3.60GHz (8 x 3600)
GPU Status|2d_canvas: enabled
GPU0: VENDOR= 0x10de, DEVICE=0x1e87 [NVIDIA GeForce RTX 2080], DRIVER_VENDOR=NVIDIA, DRIVER_VERSION=32.0.15.9186 *ACTIVE*
GPU1: VENDOR= 0x1414, DEVICE=0x008c [Microsoft Basic Render Driver], DRIVER_VERSION=10.0.26100.9061
Machine model name:
Machine model version:
direct_rendering_display_compositor: disabled_off_ok
gpu_compositing: enabled
multiple_raster_threads: enabled_on
opengl: enabled_on
rasterization: enabled
raw_draw: disabled_off_ok
skia_graphite: disabled_off
trees_in_viz: disabled_off
video_decode: enabled
video_encode: enabled
webgl: enabled
webgpu: enabled
webnn: disabled_off
Load (avg)|
Memory (System)|31.91GB (13.29GB free)
Process Argv|--folder-uri file:///e%3A/prj/planning --crash-reporter-id 5fe4547c-41ec-4de6-add1-fc92114e2670
Screen Reader|no
VM|0%
User Agent|Mozilla/5.0 (Windows NT 10.0; Win64; x64) AppleWebKit/537.36 (KHTML, like Gecko) Code/1.132.0 Chrome/148.0.7778.280 Electron/42.7.1 Safari/537.36
Installation pure|true

Extensions (4)

Name|Identifier|Author|Version
---|---|---|---
Reflow Markdown|marvhen.reflow-markdown|marvhen|3.2.0
Agency|microsoft.agency|Microsoft|1.0.260515001
rust-analyzer|rust-lang.rust-analyzer|rust-lang|0.3.2989
Svelte for VS Code|svelte.svelte-vscode|svelte|110.3.0

Theme extensions: 9

A/B Experiments

```
vsliv368:30146709
binariesv615:30325510
nativeloc1:31344060
dwcopilot:31170013
dwoutputs:31242946
copilot_t_ci:31333650
e5gg6876:31282496
pythonrdcb7:31342333
6518g693:31463988
aj953862:31281341
envsactivate1:31551504
cloudbuttont:31379625
3efgi100_wstrepl:31403338
56b7f661:31457055
ec5jj548:31422691
cmp-cht-control-diffpatch-2in1-lysithea-0014-comp:31557801
metis_rerank_treatment1:31422731
cp_cls_t_966_ss:31526232
4je02754:31466945
8hhj4413:31478653
ge8j1254_inline_auto_hint_haiku:31490510
cp_cls_t_1081:31454832
conptydll_true:31498968
core-renderer-profiling-on:31440367
e9c30283:31461165
test_treatment2:31471001
46204921:31447328
control_6dc23131:31555649
idci7584:31464702
e3e4d672:31494082
ei9d7968:31496641
nes-extended-on:31455476
chat:31457767
8hig5102:31480529
89g7j272:31518289
7e187181:31503455
i2gc6536:31499202
52612955:31516516
ghj88844:31499326
23c7c724:31520045
ddid_c:31478207
hmra_i5g22:-9
7df3h592:31512476
cp_cls_t_1082:31535311
logging_enabled_new:31498466
db5d2638:31499441
jb_cp_cls_t_632:31543129
56dj4588:31512888
32d76977:31512328
ha629193:31508444
cdk-lw-on:31524445
f9bf7985:31543584
jh5f2457_c:31540921
hgf2d445:31529595
41836166:31537719
jbcp_cls_pctr_t:31531130
61138546:31518536
3f904173:31522231
cp_intellij_t_nes:31548657
gf14b233:31526830
nes-curr-leftover-1500:31538366
ahp-both-windows:31556933
h561g132:31533256
sessions_5:31532894
ihg5j128:31534457
f412h606:31551416
7g2b5551:31542111
max-result-50:31551153
5fe6f778:31554331
allow-none:31555437
fd5dc207:31551400
c7c27ce7:31554789
nes-gb3-nf-5k:31554345
a81hg326:31551415
8hc4g144:31558143
1532g621_copy:31554320
gpt56sol_default:31557530
treatment-23-1:31555779
unuse_dynamic_mcp:31555281
a1hcc538:31559322
vrbsty_fls:31561059
session-archive:31558132
```

### Attachments