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How do I model this cylindrical extrusion with circular base?

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I first tried starting from a cylinder, although, when it comes to give thickness (by converting it to Curve/Geometry/Depth) the structure deforms and appears uneven.

I made a second endeavor by starting from a duplicated Circle, and extruding the circular vertices to match them with the other sides and finally applying a Solidify modifier but I got the following results, which are far from the object:

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Your desired mesh can easily be modeled using T-Joints. Luckily you don't have to model these because they are readily available within the built-in addon Add Mesh: Extra Objects. Enable it under menu Edit > Preferences > Add-ons and search for "Extra Objects".

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Press Shift+C to ensure the 3D Cursor is at the World Origin, and add one T-Joint object via menu Add > Mesh > Pipe Joints > Pipe T-Joint. Customize the properties of your object such as the radius within the Adjust Last Operation popup that appears in the lower left corner. Rotate the joint on the Y-Axis by 90 degrees and press Ctrl+A > Apply Rotation. Then add a Circle curve via menu Add > Curve > Circle.

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Select the pipe and add an Array Modifier and set the Count to 8.

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Then add a Curve Modifier and set the curve you added as the Curve Object.

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Now to distribute the 8 joints in an even manner, press Numpad 7 to go into top view, and adjust the circles scale and/or the Array Modifier's Factor X (or Factor Y depending on the orientation of your object) property value.

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Select the pipe objects and press tab to enter Edit Mode. Alt+LMB click one end of the pipe object to Select Edge Loop and press E to extrude and press X to extrude it along the X-Axis. Make sure the extruded edge aligns perfectly to where the end of the neighboring pipe's edge used to be aligned at. Then set the Factor X back to 1.0 so they stick side to side. You may need to frequently jump back and forth into Object/Edit Mode to view the result if the ends perfectly align on the curve. Also make sure the Array Modifier's Merge property is ticked.

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Add some Loop Cuts to the extended region to add more geometry so it can bend more smoothly.

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Once you're happy with the output you can convert the entire object to a single mesh by either applying all the modifiers or right clicking the object and selecting Convert To > Mesh. Select all the Edge Loops of the upper pipe opening and extrude it along the Z-Axis with EZ then press Shift+S > Cursor to Selected to bring up the 3D Cursor. Now tab back into Object Mode and right click the pipes and select Set Origin > Origin To 3D Cursor.

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The previous 2 steps are needed for the Mirror Modifier you will add now, and mirror it along the Z-Axis. Convert to mesh and tab into Edit Mode and press AM > Merge By Distance and increase the Merge Distance to maybe 0.001 or 0.005 to make sure all vertices have been merged. Then you can optionally add a Subdivision Surface Modifier to make it nice smooth with levels 2 or 3.

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